1
vote
3answers
46 views

Proof of $\cos \theta+\cos 2\theta+\cos 3\theta+\cdots+\cos n\theta=\frac{\sin\frac12n\theta}{\sin\frac12\theta}\cos\frac12(n+1)\theta$

State the sum of the series $z+z^2+z^3+\cdots+z^n$, for $z\neq1$. By letting $z=\cos\theta+i\sin\theta$, show that $$\cos \theta+\cos 2\theta+\cos 3\theta+\cdots+\cos ...
2
votes
2answers
52 views

Sum of a Series to Infinity

Evaluate the following sums: $\sum\limits_{i=0}^\infty\frac1{4^i}$. $\sum\limits_{i=0}^\infty\frac i{4^i}$. $\sum\limits_{i=0}^\infty\frac {i^2}{4^i}$. $\sum\limits_{i=0}^\infty\frac ...
9
votes
2answers
131 views

Summation of series $\sum_{k=0}^\infty 2^k/\binom{2k+1}{k}$

How to find the sum of this series? $$\sum_{k=0}^{\infty}\cfrac{{2}^{k}}{\binom{2k+1}{k}}$$ It seems very easy. But I still can not work it out, can anyone help?
3
votes
1answer
54 views

An equality from Representation Theory

Studying Representation Theory of finite groups I've bumped in the following identity: $$\frac{n(n+1)}{2}=\sum_{i=1}^n\frac{(2i-1)!!(2n-2i+1)!!}{(2i-2)!!(2n-2i)!!}$$ My book suggests to prove it ...
3
votes
4answers
178 views

What is $\lim_{n\to \infty}\sum_{k=1}^n \left(\frac{k}{n}\right)^n$?

I'm asked to find $\displaystyle\lim_{n\to \infty}\sum_{k=1}^n \left(\frac{k}{n}\right)^n$. It seems that the sum converges. I managed to prove that $\forall n,1<\sum_{k=1}^n ...
1
vote
3answers
55 views

Find $a_1$ given that $(1+x)^{100} = \sum_{i=0}^{100} a_ix^i$

If $(1+x)^{100} = \sum_{i=0}^{100} a_ix^i$, then $a_1$ is .. The options are $1$, $2$, $99$ or $100$. I'm sure the problem is trivial, but I just don't understand what is meant.
2
votes
1answer
53 views

How can I compute the following fast?

What approach should I adopt for computing the following problem fast? $$f(n) = \sum_{i=1}^n (n \mod i)$$ $$ 1\le n \le 10^{10}$$ Since the answer can be huge I have to output it modulo some given ...
1
vote
2answers
44 views

Closed form sum for the series given below?

Does the following series have a closed form sum? $$f(n,r) = \sum_{i=0}^n \binom{r+i}{r}$$
1
vote
0answers
35 views

Double summation of a geometric series

I am interested in the following sum for a given value of n: $ \sum\limits_{x=1}^{n} \sum\limits_{y=1}^{n}x^y$ I can simplify this to $ \sum\limits_{x=1}^{n} \frac{x^{n+1} - x}{x - 1}$ From here ...
1
vote
1answer
57 views

Summation of: $\sum_{1}^{\infty}\left(\frac{2}{3}\right)^x$ [duplicate]

This is a subsection in my statistics homework. It goes back to calculus II and summations, and it's been a long time since I've studied it so I'm rusty. I'm looking to solve the summation of ...
3
votes
0answers
147 views

How to compute this sum?

I want to sum the following: $$f(n) = \sum_{i=1}^n (i^3 \cdot (n \mod i))$$ Since the sum can be huge I have to output the sum modulo some given number m. How can I approach this problem? Also, n ...
0
votes
3answers
53 views

Find the sum $1+\cos (x)+\cos (2x)+\cos (3x)+…+\cos (n-1)x$ [duplicate]

By considering the geometric series $1+z+z^{2}+...+z^{n-1}$ where $z=\cos(\theta)+i\sin(\theta)$, show that $1+\cos(\theta)+\cos(2\theta)+\cos(3\theta)+...+\cos(n-1)\theta$ = ...
0
votes
1answer
53 views

Sum of Taylor Series

I have the converging series: $$ 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!}+... $$ and I'm trying to find its sum when x = .9. I know this is the Taylor series for some function$f(x)$, and that I can ...
2
votes
1answer
118 views

What is the infinite sum of $a^{b^x}$?

What would $$\sum^{\infty}_{n=0}(1/2)^{4^n}$$ be and how to determine it? EDIT: Apologies. I can see this converges by the ratio test. My issue is working out its sum, more for fun really. It ...
4
votes
4answers
96 views

How to derive the closed form of the sum of $kr^k$

$$ \sum_{k=0}^{n}kr^k = r\frac{1-(n+1)r^n + nr^{n+1}}{ (1 - r)^2 } $$ How to derive it? I read about some finite calculus, and i understand how to tackle sums of $x^2$, $x^3$, etc.. But I don't know ...
3
votes
2answers
123 views

Is there any closed form for the finite sum $1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+…+\dfrac{1}{n}?$ [duplicate]

I know that the infinite summation $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}+...$$ is divergent and also the sequence $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}-\ln ...
0
votes
1answer
52 views

Summation of series of $2/(r-1)(r+1)$ using the method of differences

Verify the identity $$\frac{2r-1}{r(r-1)}-\frac{2r+1}{r(r+1)}=\frac{2}{(r-1)(r+1)}$$ Hence, using the method of differences, prove that ...
1
vote
2answers
41 views

Naive proof that $\sum_{n=1}^{N-1}\cos(2\pi\frac{n}{N})=-1$ [duplicate]

As part of a larger proof, I must show that: $$\sum_{n=1}^{N-1}\cos(2\pi\frac{n}{N})=-1$$ I have thought about this but can't figure out how to get my hands on the value since I don't know any ...
17
votes
4answers
385 views

A sum containing harmonic numbers $\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n}$

I'm trying to find a closed form for the following sum $$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n},$$ where $H_n=\displaystyle\sum_{k=1}^n\frac{1}{k}$ is a harmonic number. Could you help me with it?
4
votes
3answers
92 views

How to find the sum of sequence $ 1+4+4^2+\cdots+4^{X+Y} $?

I see the following sequence and it's: $$h=1+4+4^2+\cdots+4^{X+Y}=\frac{4^{X+Y+1}-1}{4-1}$$ how we get this sequence? I know this is a primary question but I confused :)
0
votes
1answer
36 views

Prove the serie is bounded by $a_{m+1}$ for all $m\in \Bbb{N}$

Let $a_n$ a monotone sequence approaches $0$. Show that for all $m\in\Bbb{N}$: $$ 0 < (-1)^m\sum\limits_{n=m+1}^{\infty} (-1)^{n+1} {a_n} < a_{m+1} $$ I wanna focus on the RHS inequality: ...
7
votes
2answers
149 views

Sum of $1+\frac{1}{2}+\frac{1\cdot2}{2\cdot5}+\frac{1\cdot 2\cdot 3}{2\cdot 5\cdot 8}+\cdots$

I am trying to find out the sum of this $$1+\frac{1}{2}+\frac{1\cdot2}{2\cdot5}+\frac{1\cdot 2\cdot 3}{2\cdot 5\cdot 8}+\frac{1\cdot 2\cdot 3\cdot 4}{2\cdot 5\cdot 8\cdot 11}+\cdots$$. I tried with ...
7
votes
1answer
123 views

Showing that $\sum_{i=1}^n \frac{1}{|x-p_i|} \leq 8n \left( 1 + \frac{1}{3} + \frac{1}{5} + \cdots + \frac{1}{2n-1} \right)$

I'm taking a summer analysis course and preparing for our final exam later this week. Our professor gave us the following problem on our mock exam, and I can't seem to get anywhere on it. Does anyone ...
3
votes
1answer
73 views

How should I prove that: $\sum_{i=1} ^{n}(\sin(\frac{i\pi}{n}))^2=\frac{n}{2}$

$$\sum_{i=1} ^{n}\Big(\sin\big(\frac{i\pi}{n}\big)\Big)^2=\frac{n}{2}$$ An interesting conclusion and checked for validity...holds for $n\geq 2$, but yet do not know how to prove it. Are there any ...
0
votes
1answer
54 views

find the sum of the series

If $a_1, a_2, \ldots, a_n$ are in arithmetic progression whose common difference is $d$,then find the sum: $$\sin(d) \cdot \left(\csc(a_1)\csc (a_2)+\csc(a_2)\csc (a_3)+\ldots+\csc(a_{n-1})\csc(a_n) ...
2
votes
1answer
119 views

Find sum of $n$ terms of the series $12+14+24+58+164+\cdots$

Find sum of $n$ terms: $12+14+24+58+164+\cdots$ I have tried my best but could not proceed
2
votes
3answers
62 views

Find partial sums of the series $12+105+1008+10011+\dots$

Find the sum of $n$ terms of this series- $$12+105+1008+10011+.....$$ I did not understand that how should I proceed with this problem.
7
votes
0answers
70 views

Using Fourier Series to compute sums

I have just started learning the basics of Fourier series and have some doubts about it. I am aware that Fourier series can be used to compute infinite sums. For example, $\zeta(2)$ and $\eta(2)$ can ...
1
vote
2answers
50 views

Summation of $(((2N+1).2 + 1).2 + 1)\cdots $

Is there a way to sum up this series: $(((2N+1).2 + 1).2 + 1)\cdots $ The actual question that I encountered was on a coding site (HackerRank) where it said that you had a tree which grows twice in ...
5
votes
5answers
266 views

Prove that the sequence with $T(0)=1$ and $T(n) = 1 + \sum_{j=0}^{n-1}T(j)$ is given by $T(n)=2^n$

$T(0)=1 \\ T(n) = 1 + \sum_{j=0}^{n-1}T(j) \\ $ Show that $T(n) = 2^n$. I know how to prove this by induction, but I would like to know how to show this using first principles. Edit: The way I want ...
1
vote
1answer
28 views

Summation of series- substitution

If we have $\sum_{n=0}^{\infty}nf(n)=C, C\ne0\tag 1$, C is a constant, can we find a closed form for f(n)?. NB : Given condition is that $\sum_{n=0}^{\infty}f(n)$ converges to a constant value $K$ ...
-4
votes
2answers
100 views

What does this infinite sum converge to?: $\sum_{n=1}^\infty \dfrac{1}{n^k} = \dfrac{1}{1^k} + \dfrac{1}{2^k} + \dfrac{1}{3^k} + …$

$$\sum_{n=1}^\infty \dfrac{1}{n^k} = \dfrac{1}{1^k} + \dfrac{1}{2^k} + \dfrac{1}{3^k} + \dfrac{1}{4^k} + \dfrac{1}{5^k} + ...$$ I've found that: when $k=1$, it diverge to infinity when $k=2$, it ...
1
vote
2answers
64 views

Summation of infinite series

If we know the series sum given below converges to a value $C$(constant) $$\sum_{n=0}^{\infty}a_n =C \tag 2$$ Can we generate following in terms of C. values of $a_n$ will tend to zero as n goes to ...
0
votes
1answer
56 views

Brackets with a summation

$$(\sum_{i=1}^6 x_i + 9) $$ Hey guys! For above, do you think the $9$ would be part of the summation (i.e $x_1+x_2+x_3+x_4+x_5+x_6+54$) or would it be separate (i.e. $x_1+x_2+x_3+x_4+x_5+x_6+9$) ...
0
votes
3answers
29 views

Finding the sum of a sequence of terms

$$1/1(2) - 1/3(2^3) + 1/5(2^5) - 1/7(2^7)$$ This is equal to $$\sum_{n=0}^\infty(1/2)^{2n+1}(-1)^n/(2n+1)$$ Differentiating this leads to: $$\sum_{n=0}^\infty(-1/4)^n$$ Which is equal to $4/5$ Thus, ...
1
vote
7answers
118 views

Error in proving of the formula the sum of squares

Given formula $$ \sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6} $$ And I tried to prove it in that way: $$ \sum_{k=1}^n (k^2)'=2\sum_{k=1}^n k=2(\frac{n(n+1)}{2})=n^2+n $$ $$ \int (n^2+n)\ \text d ...
0
votes
2answers
62 views

Find a sum of $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{ch(n)}{3^n}$

Find a sum of $$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{ch(n)}{3^n}$$ Could you give some some hint or some way to start this? I have tried representing ch(n) through its definition with e, but I ...
4
votes
4answers
761 views

Why don't we indicate the variable to summed as we do for integrals?

When integrating over a certain variable $x$, we make sure to end the integral with $dx$, like so: $$\int_{1}^{\infty}\frac{1}{x^2}dx$$ The reason for this of course becomes more clear as one goes ...
1
vote
1answer
29 views

Transforming a power tower to a product

It is possible to write the product of a sequence of terms $a_i$ as a function of the sum of a sequence of functions of these terms: $$\prod_i a_i=f\left(\sum_i g(a_i)\right)$$ where $f=\exp$ and ...
1
vote
1answer
43 views

Random walks with finite chance of escape

In a recent answer I gave a combinatorial interpretation for the sum $\sum_{n=1} \binom{2n}{n}\frac{4^{-n}}{n+1}=1$: namely, that it corresponded to the probability of all outcomes adding to $1$. A ...
6
votes
3answers
570 views

How do you calculate this sum?

How to find the value of $S(\infty)$, where $S(n)$ is the following $$S(n)=\displaystyle\sum_{k=1}^{n} \dfrac{k}{n^2+k^2}$$ Wolfram alpha is unable to calculate it. This is a question from a ...
1
vote
1answer
70 views

Help on a tough summation from Rudin?

I'm having a tough time deriving (4) from the bracketed expression in (3) shown in the photo. I've been futzing with partial sums of geometric series and binomial expansions for a while now with no ...
7
votes
4answers
161 views

A closed form of $\sum_{k=0}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$

I am looking for a closed form of the following series \begin{equation} \mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right) \end{equation} I have no idea how to ...
1
vote
2answers
55 views

Matrix Inversion Test ( Sum of Matrix series)

Friends,I have a set of matrices of dimension $3\times3$ called $A_i$. , Following are the given conditions a) each $A_i$ is non invertible except $A_0$ because their determinant is zero. b) ...
1
vote
1answer
35 views

Sum of nth powers and generalized polynomial sum

So this is a 2-part question (both parts I believe are closely related): How exactly does on express the sum $$\sum_{i=0}^{k}{i^n} = Q(n,k)$$ in a closed form For arbitrary positive integers ...
14
votes
2answers
532 views

Intuitive ways to get formula of cubic sum

Is there an intuitive way to get cubic sum? From this post: combination of quadratic and cubic series and Wikipedia: Faulhaber formula, I get $$1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$$ I think ...
3
votes
1answer
101 views

find a $B_{n,j}$ such that $|A_{n,j}-L_j| \leq B_{n,j}$ $\forall n,j$ and $\sum_{j=0}^{\infty}B_{n,j}$ converges

We have $A_{n,j}= 3(-1)^j2^{n-j+1}\frac{(2(n-j)-4)!}{(n-j)!(n-j-2)!}\binom{j+2}{2}\frac{n^\frac{5}{2}}{8^n}$ and $L_j=(-\frac{1}{8})^j\binom{j+2}{2}\frac{3}{8\sqrt{\pi}}$ So I know $\lim_{n \to ...
11
votes
3answers
313 views

Calculate $\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$

I'm an eight-grader and I need help to answer this math problem. Problem: Calculate $$\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$$ This one is very hard for me. It ...
1
vote
1answer
88 views

combination of quadratic and cubic series

I'm an eight-grader and I need help to answer this math problem (homework). Problem: Calculate $$\frac{1^2+2^2+3^2+4^2+...+1000^2}{1^3+2^3+3^3+4^3+...+1000^3}$$ Attempt: I know how to calculate ...
3
votes
1answer
46 views

Measuring sums of complex alternating series

Suppose we have functions $$f(x) = \sqrt{x}, \space g(f) = \frac{df}{dx}+\frac{d^2f}{dx^2}+\frac{d^3f}{dx^3}\space ...$$ Applying function f(x) to g(f) we get: $$g(f(x))=\frac{1}{2}x^{-\frac{1}{2}} - ...