0
votes
2answers
45 views

Find the sum of the following series

Find the sum of the following series $$ \sum_{n=1}^\infty (-1) \frac{1}{n}\frac{9}{6^n}. $$ I think that $r$ is $\frac{9}{6^n}$ and $a$ is $-1$. But I'm not positive if I'm starting this problem ...
0
votes
2answers
16 views

General Term of specific Alternating Series

Recently I think about series below and I wonder if there is away to write the general term of it... $$ ...
4
votes
1answer
90 views

a doubt with the series $ \sum_{n=0}^{\infty}e^{-nx} $

I know that the series is equal to $$ \sum_{n=0}^{\infty}e^{-nx}= \frac{1}{1-e^{-x}}$$ However, if I expand each exponential term into a Taylor series I get $$ \sum_{n=0}^{\infty}e^{-nx}= ...
1
vote
1answer
59 views

Does the series $\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ converge?

$\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ Please let me know how you did it. Thank you.
0
votes
0answers
32 views

sequence and series of convergence problem

We have a sequence/series problem. and i need help Consider the following: How can we show that $2-2^{1/2}+2^{1/3}-2^{1/4}+2^{1/5}-2^{1/6}+\ldots$ diverges? I am so lost as to solving this problem. ...
0
votes
3answers
60 views

How does this series diverge?

The series: $$\sum_{n=0}^{\infty} \sqrt{n^2 +1} -n$$ diverges. Can someone please tell me how this is proven and done.
0
votes
3answers
41 views

Alternating p series. given that summation

Given that $$\sum_{k=1}^\infty{\frac{1}{k^2}} = \frac{\pi^2}{6}\ $$ Show that $$\sum_{k=1}^\infty{\frac{(-1)^{k+1}}{k^2}} = \frac{\pi^2}{12}\ $$
2
votes
1answer
49 views

Finding another series for a given series.

For each sequence $$a_n$$ find a number r such that $$\frac{a_n}{r^n}$$ has a finite non-zero limit. (This is of use, because by the limit comparison test the ...
0
votes
1answer
18 views

Taylor Polynomials

My question is this, Compute $T_2(x)\ $ at $x=0.8$ for $y=e^x$ I have figured out at that $T_2(x)$ equals: $$e^{0.8}+e^{0.8}(x-0.8)+\frac{e^{0.8}((x-0.8)^2)}{2}$$ The second part of the question ...
3
votes
3answers
118 views

determine if $\sum^{\infty}_{n=1} \frac{n}{e^n}$ converges

determine if $\sum^{\infty}_{n=1} \frac{n}{e^n}$ converges. I used the ratio test and eventually came to that: $\frac{n+1}{en}$ which is approximatley equal to $\frac{1}{e}$ which is less than 1. ...
1
vote
1answer
30 views

prove $\lim_{n\to\infty}\sup\{|f(x) - f_n(x) | : x \in S \} =0$

I understand how the theorem works but how would you prove that a sequence $f_n$ of functions on set $S \subset \mathbb{R}$ converges uniformly iff $$\lim_{n\to\infty} \sup\{|f(x) - f_n(x) | : x \in ...
5
votes
3answers
395 views

Is there a formal definition of convergence of series?

One is often asked to check if a given series converges or not in one's first year of uni. Is there a formal definition that allows us to check this? We are only given a bunch of tests that are ...
1
vote
2answers
38 views

Construct a specific function for a given sequence

Given the sequence $A_n=n$, I want to construct a function $f : R \to R$, such that for every $x \in N$: $f(x)=A_x$ $\int\limits_{0}^{x}f(t)dt=\sum\limits_{i=0}^{x}A_i$ How can do that? Thanks
0
votes
4answers
29 views

Writing Series as a Telescoping Series

Good evening all, I am faced with this dilemma, and I am hoping someone can help me out. The series is, $$\sum_{n=9}^{\infty}\frac{1}{n(n-1)} $$ I have figured out the sum to be $\frac{1}{8}$ but ...
0
votes
2answers
43 views

Multiplying and Dividing Series

For example, how do you compute the taylor series for $$e^x \sin x=\sum_{n=0}^{\infty} \frac {x^n}{n!} \sum_{n=0}^{\infty} (-1)^n\frac {x^{2n+1}}{(2n+1)!}$$ Of course I want the result to contain ...
0
votes
1answer
24 views

$\sum_{n=1}^{\infty}|a_{n} x^{n}| < \infty \implies \sum_{n=1}^{\infty} |a_{n}| (|x|^{n-1}+ |x^{n-2}y|+…+|y^{n-1}|) <\infty $?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$. Now, we let, $x, y \in \mathbb R$ with $x\neq ...
1
vote
3answers
43 views

$\sum_{n=1}^{\infty} a_{n}= \sum_{n=1}^{\infty}b_{n}$ and $\sum_{n=1}^{\infty} |a_{n}| < \infty \implies \sum_{n=1}^{\infty} |b_{n}| < \infty$?

Suppose that $\{a_{n}\}_{n\in \mathbb N}, \{b_{n}\}_{n\in \mathbb N} \subset \mathbb C$ such that both the series, $\sum_{n=1}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} b_{n}$ converges, and its sum ...
0
votes
1answer
69 views

Sum of a Series (Calculus )

Evaluate the following sum as $n\to\infty$: $$a_n=\sum_{k=1}^n\frac{k(k+1)}{2x^{k-1}}, \quad|x|>1$$ Source: Exercise 14, http://www.mathem.pub.ro/_SITE_ELEVI/e-2005-a1.pdf . Thank you for ...
1
vote
3answers
38 views

Comparison of two alternating series- Which is bigger

Imagine you have two finite alternating series. $$S_a=a_1-a_2+a_3-a_4+\cdots+a_n$$ $$S_b=b_1-b_2+b_3-b_4+\cdots+ b_n$$ Question: If $|a_i|>|b_i|$ is $S_a>S_b$?
0
votes
1answer
60 views

Is this construction already a contradiction

Let's say we have $\ell^p$ for $p>2$ and $\ell^2 \subsetneq \ell^p$. Let $U$ be a closed subspace of $\ell^2$. Is it possible that $U$ is isomorphic to $\ell^p$ as a Banach space?- I hardly think ...
2
votes
2answers
78 views

Convergence of $ \sum_{n=1}^{\infty} (\frac{n^2+1}{n^2+n+1})^{n^2}$

Find if the following series converge: $$\displaystyle \sum_{n=1}^{\infty} \left(\frac{n^2+1}{n^2+n+1}\right)^{n^2}$$ What I did: $$a_n=\left(\frac{n^2+1}{n^2+n+1}\right)^{n^2}$$ $$b_n=\frac ...
1
vote
1answer
50 views

Does the series $\sum\limits_{n=1}^\infty\cos (\pi/n)$ converge or diverge?

Does the series $$\sum\limits_{n=1}^\infty\cos (\pi/n)$$ converge or diverge?
1
vote
1answer
28 views

Let $ \sum_{n=1}^{\infty}a_n$ be a positive convergant series, prove that $ \sum_{n=1}^{\infty}2(a_n)^3$ converges as well

Let $ S_1=\displaystyle\sum_{n=1}^{\infty}a_n$ be a positive convergant series, prove that $ S_2=\displaystyle\sum_{n=1}^{\infty}2(a_n)^3$ converges as well. We have $\exists l :\forall ...
0
votes
1answer
34 views

Convergence and sum of geometric series (e^(3-2n)) as n goes from 2 to infinity

I have simplified the expression to: (e^3 / e^2n) This particular question asks to answer whether or not the series converges by virtue of |common ratio| < 1 alone, without using any other tests ...
0
votes
0answers
27 views

Taylor Series Calc 2

I am not sure how to find a series representation for the natural log. If anyone can show me some helpful steps to solve this problem it would be greatly appreciated. What is the Maclaurin series ...
0
votes
1answer
44 views

Test whether the series $\sum_{n = 1}^\infty (n!)^n/(n)^{4n}$ diverges or converge.

I've found this series in my calculus's book $$\sum_{n = 1}^\infty \frac{(n!)^n}{n^{4n}}$$ But the correct form is $$ \sum_{n = 1}^\infty \frac{n^n}{n^{4n}} $$ My question is: How to verify if ...
0
votes
3answers
35 views

Determine whether this series converges

I am studying for a calculus final and have come across this practice question: $\text{Determine whether the series is absolutely convergent:}$ $\sum\limits_{n=1}^{\infty} \frac{(-1)^n [1 \cdot 3 ...
1
vote
1answer
28 views

Determine the value of r where the series converges

show that $$ \big(r\big)^{ln(n)} = \big(n\big)^{ln(r)} $$ Then determine the values of r (with r>0) for which the series $$ \sum_1^\infty (\big(r\big)^{ln(n)})$$ converges. r must be in what ...
0
votes
2answers
29 views

let $f_n(x) = \frac{x^n}{1 + x^n}$. determine whether $f_n \to f$ uniformly on $[0, 1]$ and whether $f_n \to f $ uniformly on $[0, \infty)$

this sequence was given as a practice problem and I'm really having trouble. Heres the question: let $f_n(x) = \frac{x^n}{1 + x^n}$. determine whether $f_n \to f$ uniformly on $[0, 1]$ and whether ...
0
votes
4answers
51 views

Why is the derivative of circle area the circumference? [duplicate]

Why is the derivative of the volume of a sphere the surface area? And why is the derivative of the area of a circle the circumference? Too much of a coincidence, there has to be a reason! Also, why ...
3
votes
2answers
23 views

Maclaurin Series for a natural logarithm

Can anyone please help me with this question? Find the Maclaurin series and the interval of convergence for $f(x) = \ln(1-7x^9)$ I thought the answer was $$\sum_{n=1}^{\infty} (-1)^n ...
0
votes
0answers
28 views

Prove that $g_n \to g$ uniformly on $\mathbb{R}$ but $f_{n}g_n$ foes not converge uniformly to$fg$ on $\mathbb{R}$

Let $f_n(x) = x$ and $f(x) = x$ for $x \in \mathbb{R}$ and let $g_n(x) = \frac{1}{n}$ and $g(x) = 0$ for $x \in \mathbb{R}$. Prove that $f_n \to f$ uniformly on $\mathbb{R}$ and $g_n \to g$ uniformly ...
0
votes
0answers
36 views

Total Time for a ball that bounces from a height of 8 feet and rebounds to a height 5/8 [duplicate]

I am trying to solve the following problem. Let's say a ball is dropped from a height of $8$ feet and rebounds to a height $\frac{5}{8}$ of its previous height at each bounce keeps bouncing ...
0
votes
2answers
65 views

Showing $a_n=\sin(n)$ does not converge

Show that $a_n=\sin(n)$ does not converge My idea: Take two subsequences: $a_{n_k}=\sin(\frac {\pi k} 2)$ , $a_{n_l}=\sin(\frac {2\pi l} 3)$ So: $\forall n$ : $\lim_{n\to\infty} a_{n_k}=1$, ...
0
votes
2answers
69 views

Total Time a ball bounces for from a height of 8 feet and rebounds to a height 5/8

I am trying to solve the following problem. Let's say a ball is dropped from a height of $8$ feet and rebounds to a height $\frac{5}{8}$ of its previous height at each bounce keeps bouncing ...
0
votes
2answers
30 views

Taylor Series Maclaurin Series Interval Expansion

Hi! I am currently woking on some clack online homework problem. I really have no idea how to approach this problem. If someone could help me solve this question I would greatly appreciate it!
0
votes
2answers
49 views

$l^r \subset l^p$ and is it even a subspace

It is true that for $r<p$ and $r,p \in [1,\infty)$ we have that $l^r \subset l^p$. Is it true that $l^r$ cannot be isomorphic to a subspace of $l^p$?
1
vote
3answers
82 views

Convergence of series minus logarithm

im trying to solve this problem since two, three days.. Is there someone who can help me to solve this problem step by step. I really want to understand & solve this! $$ Show\ \exists \ \beta ...
5
votes
2answers
87 views

Convergence of $S_n(a) = \sum_{an < k \leqslant (a+1)n} \frac{1}{\sqrt{kn-an^2}}$

Let $a\in\mathbb{R}$ et $n \in\mathbb{N}$, Denote the following sequence, $$\displaystyle S_n(a) = \sum_{an < k \leqslant (a+1)n} \frac{1}{\sqrt{kn-an^2}}$$ For which values ​​of $a$ the ...
2
votes
2answers
40 views

Proving a sequence to be divergent

I'm trying to prove this sequence: $a_n = \sqrt{n}-\sqrt{n^2-1}$ to be divergent. How would I do this? I'm thinking of proving that it's not bounded below, but I'm not sure how to do that with ...
4
votes
2answers
120 views

Doubly infinite sequence limit

I have a 2-indexed sequence $F^n_m$ where $n,m$ are natural numbers and I am concerned about the behavior as $n\to\infty$ and/or $m\to\infty$. The sequence is expressed as ...
5
votes
1answer
103 views

Show that: $\sum {\min \left( {{a_n},{1 \over n}} \right)} = \infty $

Let $a_n$ be a sequence decreasing to $0$, and $\sum {{a_n} = \infty } $. Show that:$$\sum {\min \left( {{a_n},{1 \over n}} \right)} = \infty $$ If there's $N_0$ such that: $\forall n>N_0: ...
0
votes
0answers
45 views

How do I begin to prove this infinite series equation with positive integers? [duplicate]

How do I begin to prove this infinite series equation with positive integers? I want to know what the beginning steps are, or ideas to consider when trying to prove this: $$\lim_{k \to ...
0
votes
2answers
68 views

Does this series converge? $\sum_{k=1}^{\infty}\frac{1}{\sqrt{k}}\sin^2\frac{1 }{\sqrt{k}}$?

What should I do with such problems? $\sum_{k=1}^{\infty}\frac{1}{\sqrt{k}}\sin^2\frac{1 }{\sqrt{k}}$ Thanks!
1
vote
4answers
513 views

Series is convergent but it seems it is divergent?

I have a series: $$ \sum^\infty_{n=1}{\bigg(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+2}}\bigg)} $$ and I thought it is a divergent series since $$ \sum{\big(f(x)-g(x)\big)} = \sum{f(x)} - \sum{g(x)} $$ ...
1
vote
2answers
43 views

Bounding $\sum_{n=n_1}^\infty x^n (n+1)^2$

I need to upperbound the sum $$\sum_{n=n_1}^\infty x^n (n+1)^2$$ where $0<x<1$ is a parameter. I know it can be done starting from $$\sum_{n=n_1}^\infty x^n (n+1)^2\le \sum_{n=0}^\infty x^n ...
0
votes
1answer
26 views

Maclaurin Series multiplying in a constant

So I understand how to set up this series but I'm just confused on the last part so the question is find the maclaurin series for the following: $$f(x) = 15x \cos \left( \frac{1}{14}x^2 \right)$$ so ...
3
votes
1answer
43 views

Alternating series limit question [closed]

Suppose $b_n > 0$ for all $n\geq1$ and $$\lim_{n\to\infty}n\left(\frac{b_n}{b_{n+1}}-1\right)>0,$$ show that the alternating series $\sum_{n=1}^\infty(-1)^n b_n$ converges.
8
votes
4answers
219 views

Find the infinite sum of the series $\sum_{n=1}^\infty \frac{1}{n^2 +1}$

This is a homework question whereby I am supposed to evaluate: $$\sum_{n=1}^\infty \frac{1}{n^2 +1}$$ Wolfram Alpha outputs the answer as $$\frac{1}{2}(\pi \coth(\pi) - 1)$$ But I have no idea ...
0
votes
0answers
21 views

Convergent series and sum - Proof validation

Let $u_0 > 0$ be a real number and let $a_n$ be a sequence of strictly positive real numbers. Let $$u_{n+1} = u_n + \frac{a_n}{u_n}$$ Show that $u_n$ is convergent if and only if $\sum a_n < ...