0
votes
0answers
7 views

Radius and Interval of Convergence of $\sum\limits_{n=1}^{\infty}\dfrac{x^n}{2n-1}$

This is my first time finding the radius and interval of convergence of a series, so please bear with me. I would like to find the radius and interval of convergence of ...
1
vote
0answers
17 views

Closed form of $\sum_{k=1}^{\infty }\left(\psi_1(k)\right)^n$

Inspired by answers to this question, for which $n$ values could we specify a closed-form of $$S(n)=\sum_{k=1}^{\infty }\left(\psi_1(k)\right)^n\,?$$ Here $\psi_1$ is the trigamma function, and ...
0
votes
1answer
21 views

Studying the convergence of a series with logarithm [on hold]

I would like to know if this sum is convergent, and why: $$\sum_1^\infty\ln\bigg(1+\dfrac1{n^a}\bigg)$$
0
votes
5answers
51 views

Convergence/Divergence of the series $\sum\limits_{n=1}^{\infty}\tan(1/n)$

Trying to see if $$\sum\limits_{n=1}^{\infty}\tan(1/n)$$ converges or diverges. As $n \to \infty$, $\tan(1/n) \to 0$, so inconclusive. Ratio test was inconclusive, root test doesn't look good for this ...
0
votes
2answers
29 views

Convergence of the series $\sum\limits_{n=1}^{\infty}(-1)^{n}\dfrac{\ln(n)}{\sqrt{n}}$

I would like to see whether or not $$\sum\limits_{n=1}^{\infty}(-1)^{n}\dfrac{\ln(n)}{\sqrt{n}}$$ is a convergent series. Root test and ratio test are both inconclusive. I tried the alternating ...
0
votes
3answers
50 views

Showing the divergence of the series where $a_1 = 2$ and $a_{n+1} = \frac{5n+1}{4n+3}a_n$.

Consider a series such that its $i$th term $a_i$ is defined by $a_1 = 2$ and $a_{n+1} = \dfrac{5n+1}{4n+3}a_n$. I would like to show that this series is divergent. Here's how I thought about it: ...
1
vote
0answers
26 views

Prove that the relationship exists

This question sprung out from another post of mine that was in part by Semiclassical, he Proved the Following: $$ \sum_{n=0}^{\infty} {}_2F_1(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};1)/n! = 2\pi ...
4
votes
2answers
111 views

The Series $-\frac{2}{5}+\frac{4}{6}-\frac{6}{7}+\frac{8}{8}-\frac{10}{9}-\cdots$

Stewart claims that this series is convergent, but Wolfram and I disagree. I looked at $$\lim\limits_{k\to\infty}\dfrac{(-1)^k (2k)}{4+k} $$ which is clearly not 0. Did I do something wrong?
3
votes
1answer
48 views

Convergence of $\sum x_k \log (x_k^{-1}) $ [on hold]

Prove or disprove: If $\{x_k\} \subset (0,1)$, and $\sum x_k < \infty$, then $\sum x_k \log (x_k^{-1}) < \infty$. What if $\{x_k\}$ is monotone? Thanks a lot.
1
vote
0answers
28 views

Question about series and how the pattern idea works

Two Questions: When you are given: $1, 2, 3, .... , n$ How do you know that in the $...$ that it continues the $x_{n-1} + 1$ pattern? Is it the definition of series? Secondly: Do partial sums ...
2
votes
0answers
32 views

Prove that $\lim_{n\to\infty} \left( \frac{ g_n^{\gamma}}{\gamma^{g_n}} \right)^{2n} = \frac{e}{\gamma}$

Put $g_n = 1 + \frac{1}{2} + ... + \frac{1}{n} - \log(n)$. Prove that $$\lim_{n\to\infty} \left( \frac{ g_n^{\gamma}}{\gamma^{g_n}} \right)^{2n} = \frac{e}{\gamma}$$ I've tried this for a while now ...
3
votes
4answers
411 views

How to show this series diverges

How to show the following series $$\sum_{n=2}^\infty \frac{1}{n \log n}$$ diverge? Thank you!
0
votes
0answers
40 views

which of the following is sufficient for $\displaystyle \lim_{n\to \infty} \frac{a_n}{b_n}=1$?

Suppose sequences $a_n,b_n$ both have limits and are finite then which of the following is sufficient for $\lim_{n\to \infty} \frac{a_n}{b_n}=1$? $1.\lim_{n\to\infty} a_n=\lim_{n\to \infty} b_n $ ...
5
votes
3answers
105 views

Is there any method to get a finite sum for $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots$?

As we can see on Wikipedia, there are some algebraic methods that give us finite sums for the Grandi's series $$1-1+1-1+1-1+1-1+\cdots$$ Let $S$ be the sum of the Grandi's series. Then ...
2
votes
0answers
39 views

Time to buy a house without a mortgage equation!!

I am looking into a "real world" calculation to calculate the time taken for someone to buy their own home while they rent it. They do this by buying small pieces of the property every month, and ...
5
votes
2answers
98 views

Evaluating a series of hypergeometric functions

I would like to prove (or disprove) the following statement: $$ \sum_{n=0}^\infty \left[\frac{{}_2{\rm F}_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};1\right)}{n!}\right] = \frac{\pi}{2} \left[ ...
1
vote
2answers
42 views

Determine whether or not $\sum_{k=1}^{\infty} \frac{1}{k- \mathrm{e}^{-k}}$ converges.

I have the following so far Let $a_k = \frac{1}{k- e^{-k}}$. Now, $\lim_{k \to \infty}a_k=0 \implies$ $\sum a_k$ can either converge or diverege. We must thus do further tests to determine whether ...
2
votes
1answer
36 views

Suppose the limit of $f(z)$ as $z$ approaches $z_0$, exists and call it $w_0$. Suppose a sequence $(a_n)$ converges to $z_0$. Does $f(a_n)$ converge

Suppose the limit of $f(z)$ as $z$ approaches $z_0$, exists and call it $w_0$. Suppose a sequence $(a_n)$ converges to $z_0$. Does $f(a_n)$ converge to $w_0$ and $n \rightarrow \infty$? I would say ...
2
votes
2answers
15 views

In $(]0,1], d_{\mid.\mid})$ why $(\frac{1}{1+n})_{n\in \mathbb{N}}$ do not converge?

Can someone tell me why this sequence do not converge ? First, I know that is a Cauchy's sequence. Then, the fact is that the sequence converges to $0$ when $n \rightarrow \infty$. Thanks in ...
0
votes
2answers
36 views

How to estimate $\sum_{n=0}^\infty (\log\log2)^n/n! $ from below?

How to show that the following inequality holds: $$\sum_{n=0}^\infty \frac{(\log\log2)^n}{n!}>\frac 35$$ Is it possible to prove this using induction?
0
votes
0answers
26 views

Analysis .Find sequences? [on hold]

Find sequences $a_n, b_n$ such that \begin{align*} &a_n=\frac{x_n}{y_n} 3^{1/2}, \quad a_n \rightarrow 1 \\ &b_n = \frac{z_n}{w_n}, \quad b_n \rightarrow 3^{\frac{1}{2}} \end{align*} where ...
1
vote
0answers
101 views

Prove $\sum_{n=1}^\infty \sum_{m=2^{n+1}} ^\infty \frac{(-1)^mn}{m}$ converges and evaluate it.

Noting that the series inside is a part of the alternating harmonic series multiplied by $-n$, we get $\displaystyle \sum_{m=2^{n+1}} ^\infty \frac{(-1)^mn}{m}=\sum_{m=1} ^\infty \frac{(-1)^mn}{m} ...
1
vote
1answer
41 views

Finding General Term of a Repititive Sequence

I'm doing my mathematics homework and there's question which I'm pretty much unable to solve. I've literally tried every method but no results. It'd be great, if anybody could help me. Thanks (in ...
1
vote
1answer
37 views

Evaluating $ \sum_{i=0}^{\infty}ir^i$ for $|r|\lt 1$ [duplicate]

I'm doing my mathematics homework and there's question which I'm pretty much unable to solve. I've literally tried every method but no results. It'd be great, if anybody could help me. Thanks (in ...
2
votes
0answers
36 views

How to calculate alternating Euler sum [on hold]

In How to calculate $\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$? get $$\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2=\frac{1}{12}(\pi^2\log2-4(\log 2)^3-9\zeta(3)),$$ Similar, how to evaluate the series, ...
5
votes
3answers
121 views

Some integral representations of the Euler–Mascheroni constant

What kind of substitution should I use to obtain the following integrals? $$\begin{align} \int_0^1 \ln \ln \left(\frac{1}{x}\right)\,dx &=\int_0^\infty e^{-x} \ln x\,dx\tag1\\ &=\int_0^\infty ...
0
votes
2answers
39 views

How to decide about the convergence of $\sum(n\log n\log\log n)^{-1}$?

In Baby Rudin, Theorem 3.27 on page 61 reads the following: Suppose $a_1 \geq a_2 \geq a_3 \geq \cdots \geq 0$. Then the seires $\sum_{n=1}^\infty a_n$ converges if and only if the series $$ ...
2
votes
1answer
25 views

How should Theorem 3.22 in Baby Rudin be modified so as to yield Theorem 3.23 as a special case?

I'm reading Walter Rudin's PRINCIPLES OF MATHEMATICAL ANALYSIS, 3rd edition, and am at Theorem 3.22. Theorem 3.22: Let $\{a_n\}$ be a sequence of complex numbers. Then $\sum a_n$ converges if and ...
1
vote
5answers
194 views

Why are Maclaurin series useful if we can only use them for such a small range of numbers?

Okay, I am beginning to get how Maclaurin series work, but what I don't understand is why they are useful. Why would you want an infinite expansion for a series that works for such few values (only ...
4
votes
2answers
97 views
+50

Closed form of $\sum_{k=0}^{\infty} \frac{k^a\,b^k}{k!}$

While working on this question I think I've found a closed-form expression for the following series, but I don't know how to prove it. Let $a \in \mathbb{N}$ and $b \in \mathbb{R}$. Then ...
6
votes
0answers
217 views
+100

It is easy to show that $\displaystyle S_m=\sum_{n=1}^\infty \frac{n}{2^n + m}$ converges for any natural$\ m$, but what is its value?

In fact the series would converge even if$\ m$ were not natural, I just wanted to state that it is natural in my case. I have found the partial sum formula of$\ S_0$,$\displaystyle \sum_{n=1}^k ...
1
vote
1answer
35 views

Can anyone please clarify for me certain points I have in the proof of Theorem 3.17 in Baby Rudin?

I'm currently studying, or revising, Walter Rudin's Principles of Mathematical Analysis, 3rd eition, and I'm stuck with the proof of Theorem 3.17. Here's the statement of the Theorem: Let ...
2
votes
2answers
59 views

Closed form of the sequence ${_2F_1}\left(\begin{array}c\tfrac12,-n\\\tfrac32\end{array}\middle|\,\frac{1}{2}\right)$

Is there a closed-form of the following sequence? $$a_n={_2F_1}\left(\begin{array}c\tfrac12,-n\\\tfrac32\end{array}\middle|\,\frac{1}{2}\right),$$ where $_2F_1$ is the hypergeometric function and $n ...
0
votes
2answers
28 views

How to state convergence through limit comparison test?

I am able to show convergence of the following series through the root test but am trying to practice the limit comparison test and can't figure out how to do it that way. $$\sum_{n=1}^\infty ...
1
vote
1answer
89 views

Is the series: $\,\sum_{n=1}^{\infty}\frac{\mathrm{e}^{-in}}{n}\,$ divergent?

According to mathematica, the complex series $\displaystyle\sum_{n=1}^{\infty}\frac{e^{-in}}{n}$ does not converge. I know that the factor $\dfrac{1}{n}$ in the above series is diverging, but I don't ...
0
votes
1answer
27 views

Calc I limit/series question

Let $f : \mathbb R\rightarrow\mathbb R$ be a function that is differentiable at zero and such that $f(0)=0$. Show that for each $n\in \mathbb N$ we have that ...
2
votes
1answer
26 views

How are these definitions of the limit superior and limit inferior equivalent?

I have come across these three definitions of the limit superior (or upper limit) and the limit inferior (or lower limit) of a sequence of real numbers and I wonder how to establish the equivalence of ...
2
votes
2answers
65 views

Question on simplification of $\sum_{n=1}^{\infty}\frac{2}{(2n+1)(2n+3)}$?

I am having trouble seeing how $\sum_{n=1}^{\infty}\frac{2}{(2n+1)(2n+3)}$ equals $\sum_{n=1}^{\infty}\frac{1}{2n+1}-\frac{1}{2n+3}$. I can see $\sum_{n=1}^{\infty}\frac{1}{2n+1}+\frac{1}{2n+3}$ but ...
0
votes
5answers
49 views

Convergence of $\sum_{n=1}^\infty \frac{\sqrt{n+1}-\sqrt{n}}{n^p} $

Investigate the convergence of $\displaystyle\sum_1^\infty \frac{\sqrt{n+1}-\sqrt{n}}{n^p}$. For what values of p does the series converge? I have applied the ratio test, which is inconclusive. Would ...
0
votes
2answers
37 views

The convergence of sum $\sum{\frac {(N^3 + 2n^5 -n^6)}{(n^2+n^4+2n^8)}}$

The convergence of series $$\sum_{n=1}^{\infty}{\frac {(n^3 + 2n^5 -n^6)}{(n^2+n^4+2n^8)}}$$Preferably with the comparison test.
1
vote
3answers
30 views

Compute the following sum for any x?

Compute the following sum for any x? $\sum_{n=0}^\infty {(x-1)^n\over (n+2)!}$ I am having trouble to compute that sum. It looks like geometric series but I don't know where to start. Can everyone ...
2
votes
0answers
147 views
+200

Infinite Series -: $\psi(s)=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+.+.+ $.

We have a given converging series using derivatives and matrices(Analogue to Taylor's series) $\psi(s)_{3 \times 3}=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+..+.. \tag 1$. ...
0
votes
3answers
37 views

convergente of the sum of sines of the terms of the alternating harmonic series

I want to know about the convergence or divergence of the following series: $$\sum \sin (a_n) $$ where $$a_n=\frac{(-1)^n}{n}$$ The tests that I tried were inconclusive. Is it possible to know? ...
15
votes
2answers
145 views

Limit of $\int_0^1\frac{B_{2n+1}\left(\left\{\frac1x\right\}\right)}{x}dx$

Set $$u_n= \int_0^1 \frac{B_{2n+1}\left(\left\{\frac1x\right\}\right)}{x}dx\tag1$$ where $\left\{t\right\}=t-\lfloor t \rfloor$ denotes the fractional part of $t$ and where $B_n(\cdot)$ are the ...
1
vote
1answer
17 views

Sequence increase/decrease

I need help determining if the following sequences are increasing, decreasing or not monotnic. Any help would be great. I think that for c) it is not monotonic as well as for a. a) ...
0
votes
2answers
95 views

Show that $\,a_n=f(1)+f(2)+\cdots+f(n)-\int_1^n f(x)\,dx\,\,$ converges

Let $\,f:[1,\infty)\to \mathbb R\,$ be a decreasing and lower bounded function. Show that the sequence $\{a_{n}\}_{n\in\mathbb N}$ defined as: $$ a_n=f(1)+f(2)+\cdots+f(n)-\!\int_1^n\!\! f(x)\,dx, $$ ...
1
vote
1answer
34 views

How to obtain this geometric progression

How do I obtain this from the formula of the geometric progression (which I 'only' know as $1+q+q^2+...+q^{n-1} = \frac{1-q^n}{1-q}$)? $$\frac{x_1^p-x^p}{x_1^q-x^q} = ...
4
votes
3answers
95 views

Show that $\lim_{n \to \infty} \frac{1}{n!} \frac{n^{n+1}}{e^n} = \infty$

Show that $$\lim_{n \to \infty} \frac{1}{n!} \frac{n^{n+1}}{e^n} = \infty$$ Wihout using stirlings aproximation to n! I've tried to compare this to a divergent sequence but didnt work. Also, I dont ...
2
votes
1answer
29 views

Verification of identity $2\sum_{m,n\geq 1\,;\,m>n}\frac{1}{m^k n^k}=\left(\sum_{n\geq 1}\frac{1}{n^k}\right)^2 -\sum_{n\geq 1}\frac{1}{n^{2k}}$

Is this identity true? $$2\sum_{m,n\geq 1\,;\,m>n}\frac{1}{m^k n^k}=\left(\sum_{n\geq 1}\frac{1}{n^k}\right)^2 -\sum_{n\geq 1}\frac{1}{n^{2k}}$$ If so, how to prove it? Could you provide me a ...
8
votes
2answers
112 views

Prove that $\sum_{k=0}^\infty \frac{1}{16^k} \left(\frac{120k^2 + 151k + 47}{512k^4 + 1024k^3 + 712k^2 + 194k + 15}\right) = \pi$

How to prove the following identity $$\sum_{k=0}^\infty \frac{1}{16^k} \left(\frac{120k^2 + 151k + 47}{512k^4 + 1024k^3 + 712k^2 + 194k + 15}\right) = \pi$$ I am totally clueless in this one. Would ...