Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.

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3
votes
4answers
2k views

Proving the sum of the first $n$ natural numbers by induction

I am currently studying proving by induction but I am faced with a problem. I need to solve by induction the following question. $$1+2+3+\ldots+n=\frac{1}{2}n(n+1)$$ for all $n > 1$. Any ...
0
votes
2answers
881 views

Summation of series of product of Fibonacci numbers

What is the sum of following product of Fibonacci numbers $$\sum_{k=1}^{n-1} Fib(k)*Fib(n+3-k)$$ can anyone suggest only approach to find general term?
0
votes
5answers
160 views

Prove by induction: $2^n = C(n,0) + C(n,1) + \cdots + C(n,n)$ [duplicate]

This is a question I came across in an old midterm and I'm not sure how to do it. Any help is appreciated. $$2^n = C(n,0) + C(n,1) + \cdots + C(n,n).$$ Prove this statement is true for all $n ...
17
votes
3answers
426 views

How prove this sum $\sum_{n=1}^{\infty}\binom{2n}{n}\frac{(-1)^{n-1}H_{n+1}}{4^n(n+1)}$

show that $$\sum_{n=1}^{\infty}\binom{2n}{n}\dfrac{(-1)^{n-1}H_{n+1}}{4^n(n+1)}=5+4\sqrt{2}\left(\log{\dfrac{2\sqrt{2}}{1+\sqrt{2}}}-1\right)$$ where ...
13
votes
3answers
323 views

$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{(-1)^m E_{2m}\pi^{2m+1}}{4^{m+1}(2m)!}$

I'm looking for a way to prove $$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{(-1)^m E_{2m}\pi^{2m+1}}{4^{m+1}(2m)!}$$ Since ...
13
votes
3answers
361 views

Show that $\sum_{k=0}^n\binom{2n}{2k}^{\!2}-\sum_{k=0}^{n-1}\binom{2n}{2k+1}^{\!2}=(-1)^n\binom{2n}{n}$

How can I prove the identity: $$ \sum_{k=0}^n\binom{2n}{2k}^2-\sum_{k=0}^{n-1}\binom{2n}{2k+1}^2=(-1)^n\binom{2n}{n}? $$ Maybe, can we expand $$ f(x)=(1+x)^{2n}? $$ Thank you.
12
votes
6answers
886 views

Proving a binomial sum identity

Mathematica tells me that $$\sum _{k=0}^n { n \choose k} \frac{(-1)^k}{2k+1} = \frac{(2n)!!}{(2n+1)!!}.$$ Although I have not been able to come up with a proof. Proofs, hints, or references are all ...
9
votes
2answers
150 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?
6
votes
3answers
2k views

Solve $\sum nx^n$

I am trying to find a closed form solution for $\sum_{n\ge0} nx^n\text{, where }\lvert x \rvert<1$. This solution makes sense to me: $\sum_{n\ge0} x^n=(1-x)^{-1} \\ \frac{d}{d x} \sum_{n\ge0} x^n ...
5
votes
3answers
1k views

Proving that $\sum_{k=0}^{n} {{m+k} \choose{m}} = { m+n+1 \choose m+1 }$

I have to prove that: $$\sum_{k=0}^{n} {{m+k} \choose{m}} = { m+n+1 \choose m+1 }$$ I tried to open up the right side with Pascal's definition that: $$ { n \choose k} = {n-1 \choose {k}} + {n-1 ...
4
votes
1answer
142 views

Finite Series - reciprocals of sines

Find the sum of the finite series $$\sum _{k=1}^{k=89} \frac{1}{\sin(k^{\circ})\sin((k+1)^{\circ})}$$ This problem was asked in a test in my school. The answer seems to be ...
2
votes
3answers
7k views

How many distinct functions can be defined from set A to B

In my discrete mathematics class our notes say that between set A (having 6 elements) and set b (having 8 elements), there are 8^6 distinct functions that can be formed, in other words: |b|^|a| ...
6
votes
3answers
643 views

How do I evaluate this sum(involving the floor function)?

$$ \sum_{i=1}^N\left\lfloor\frac{N}{i}\right\rfloor $$ Is there a closed form expression to the above sum? (Mathematica doesn't give me anything)
21
votes
4answers
528 views

How find this sum $I_n=\sum_{k=0}^{n}\frac{H_{k+1}H_{n-k+1}}{k+2}$

$$I_n=\sum_{k=0}^{n}\dfrac{H_{k+1}H_{n-k+1}}{k+2}$$ where $$H_{n}=1+\dfrac{1}{2}+\cdots+\dfrac{1}{n}$$ my try:since ...
21
votes
1answer
610 views

Simplify $\left({\sum_{k=1}^{2499}\sqrt{10+{\sqrt{50+\sqrt{k}}}}}\right)\left({\sum_{k=1}^{2499}\sqrt{10-{\sqrt{50+\sqrt{k}}}}}\right)^{-1}$

Simplify $$\frac{\displaystyle\sum_{k=1}^{2499}\sqrt{10+{\sqrt{50+\sqrt{k}}}}}{\displaystyle\sum_{k=1}^{2499}\sqrt{10-{\sqrt{50+\sqrt{k}}}}}$$ I don't have any good idea. I need your help.
20
votes
1answer
831 views

Solving $x^k+(x+1)^k+(x+2)^k+\cdots+(x+k-1)^k=(x+k)^k$ for $k\in\mathbb N$

Letting $k$ be a natural number, can we solve the following $k$-th degree equation ? $$x^k+(x+1)^k+(x+2)^k+\cdots+(x+k-1)^k=(x+k)^k\ \ \ \ \cdots(\star).$$ The following two are famous: ...
7
votes
6answers
254 views

Finding $\binom{n}{0} + \binom{n}{3} + \binom{n}{6} + \ldots $

Help me to simplify:$$\binom{n}{0} + \binom{n}{3} + \binom{n}{6} + \ldots $$ I got a hunch that it will depend on whether $n$ is a multiple of $6$ and equals to $\frac{2^n+2}{3}$ when $n$ is a ...
16
votes
4answers
420 views

Proving $\sum_{k=0}^{2m}(-1)^k{\binom{2m}{k}}^3=(-1)^m\binom{2m}{m}\binom{3m}{m}$ (Dixon's identity)

I found the following formula in a book without any proof: $$\sum_{k=0}^{2m}(-1)^k{\binom{2m}{k}}^3=(-1)^m\binom{2m}{m}\binom{3m}{m}.$$ I don't know how to prove this at all. Could you show me how ...
11
votes
5answers
436 views

Combinatorial interpretation of an alternating binomial sum

Let $n$ be a fixed natural number. I have reason to believe that $$\sum_{i=k}^n (-1)^{i-k} \binom{i}{k} \binom{n+1}{i+1}=1$$ for all $0\leq k \leq n.$ However I can not prove this. Any method to prove ...
14
votes
1answer
1k views

Combinatorial proof of a Fibonacci identity: $n F_1 + (n-1)F_2 + \cdots + F_n = F_{n+4} - n - 3.$

Does anyone know a combinatorial proof of the following identity, where $F_n$ is the $n$th Fibonacci number? $$n F_1 + (n-1)F_2 + \cdots + F_n = F_{n+4} - n - 3$$ It's not in the place I thought it ...
7
votes
1answer
191 views

Sum $\sum^\infty_{n=1}\frac{(-1)^nH_n}{(2n+1)^2}$

I would like to seek your assistance in computing the sum $$\sum^\infty_{n=1}\frac{(-1)^nH_n}{(2n+1)^2}$$ I am stumped by this sum. I have tried summing the residues of $\displaystyle ...
6
votes
2answers
127 views

How find this $\sum_{n=1}^{\infty}\frac{(-1)^{n-1}\zeta_{n}(3)}{n}=?$

Question: show that $$\sum_{n=1}^{\infty}\dfrac{(-1)^{n-1}\zeta_{n}(3)}{n}=\dfrac{19\pi^4}{1440}-\dfrac{3}{4}\zeta{(3)}\ln{2}?$$ where $$\zeta_{n}(3)=\sum_{k=1}^{n}\dfrac{1}{k^3}$$ But I use ...
10
votes
7answers
403 views

Why does $\sum_{k=1}^{\infty}\dfrac{{\sin(k)}}{k}={\dfrac{\pi-1}{2}}$?

Inspired by this question (and far more straightforward, I am guessing), Mathematica tells us that $$\sum_{k=1}^{\infty}\dfrac{{\sin(k)}}{k}$$ converges to $\dfrac{\pi-1}{2}$. Presumably, this can ...
11
votes
3answers
776 views

Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.

Prove that $$\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$$ I can't find counting interpretations for either of the sides. A hint of "if $S$ is a subset of $\{1, . . . , n\}$ and $S^\prime$ is its ...
0
votes
3answers
221 views

$1+2+3+4+5+… = -\frac{1}{12}$. Is there any intuition for this? [duplicate]

I was looking into a Numberphile video here. The guy says he was unable to find an intuition. Does there exist one? Is the premise, $1-1+1-1+...=\frac{1}{2}$, reasonable mathematically?
0
votes
2answers
270 views

How to simplify this summation containing floor

Here's the summation: $$ S = \sum_{i=1}^{n}\left\lfloor\frac{n}{i}\right\rfloor $$ It can also be written as $$ S = \sum_{i=1}^{n}\left\lfloor\frac{n- (n \mod i)}{i}\right\rfloor $$ The answer will ...
5
votes
1answer
401 views

Integrating $\sum_{n=1}^{\infty} \frac{\sin nx }{n^4}$

Consider : $\displaystyle f(x)= \sum_{n=1}^{\infty} \frac{\sin nx }{n^4}$ Find : $\displaystyle \int_0^{x} f(t)\ \mathrm{d}t$.
3
votes
1answer
169 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 ...
3
votes
1answer
267 views

Evaluate $\sum_{k = 0}^{n} {n\choose k} k^m$

So, I wonder what is the evaluation of $$\sum_{k = 0}^{n} {n\choose k} k^m\text{,}\qquad (*)$$ where $m,n\in \mathbb{N}$. One of my tries: knowing that $$k^m = \sum_{j = 0}^{m}\text{S}(m,j)\cdot ...
3
votes
5answers
382 views

Evaluating even binomial coefficients

Can someone give me a hint how to evaluate $$ \binom{n}{0}+\binom{n}{2}+\cdots+\binom{n}{o(n)},$$ where $o(n)$ is $n$ if $n$ is even and $n-1$ otherwise ?
2
votes
2answers
255 views

Double Summation: Need help to handle $ i \neq j $ : $ \sum_{i=0 \to 7,\ j=1 \to 8,\ i\neq j} (8i + j) $

[Q1]. Can I ? ( write the same summation as ) : $$ \sum_{i=0, i \neq j}^7 \sum_{j=1}^8 (8i + j) \tag{1}$$ I tried to solve the following Summation as follows: Let i = m-1 then, $ \sum_{i=0,\ i ...
2
votes
3answers
111 views

How would you prove $\sum_{i=1}^{n} (3/4^i) < 1$ by induction?

How would you prove this by induction? $\sum_{i=1}^{n} (3/4^i) < 1$, $ \forall n \geq 2$ I can do the base case but don't know how to to finish it
1
vote
3answers
1k views

Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$

I'm repeating material for test and I came across the example that I can not do. How to calculate this sum: $\displaystyle\sum_{k=0}^{n}{2n\choose 2k}$?
0
votes
1answer
170 views

Is every sum to something a limit of the sum?

Old title: Is it always true that a sum to some $m$ equals the limit of that sum to some $x$ such that $x$ tends to $m$? $\ $ This may seem a stupid question, but I'm placing it because although ...
-1
votes
1answer
104 views

Use Mathematical Induction to prove that $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} +…+\frac{1}{n(n+1)}=1-\frac{1}{n+1}$

Use Mathematical induction to prove that for all integers, $n$ is greater than or equal to $1$. I am confused on what to do after I do the the basis step that is using $n$ as $1$. $$\frac{1}{1 \cdot ...
32
votes
4answers
671 views

$\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor$ is true?

I found the following relational expression by using computer: For any natural number $n$, $$\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor.$$ Note ...
29
votes
1answer
552 views

Geometric interpretation for sum of fourth powers

Summing the first $n$ first powers of natural numbers: $$\sum_{k=1}^nk=\frac12n(n+1)$$ and there is a geometric proof involving two copies of a 2D representation of $(1+2+\cdots+n)$ that form a ...
28
votes
6answers
1k views

Evaluating $‎\sum_{n=2}^{\infty}\frac{\zeta(n)}{k^n}$

‎If $f\left(z \right)=\sum_{n=2}^{\infty}a_{n}z^n$ and $\sum_{n=2}^{\infty}|a_n|$ converges then‎, $$\sum_{n=1}^{\infty}f\left(\frac{1}{n}\right)=\sum_{n=2}^{\infty}a_n\zeta\left(n\right)‎.$$ ‎Since ...
37
votes
1answer
791 views

Why does this ratio of sums of square roots equal $1+\sqrt2+\sqrt{4+2\sqrt2}=\cot\frac\pi{16}$ for any natural number $n$?

Why is the following function $f(n)$ constant for any natural number $n$? $$f(n)=\frac{\sum_{k=1}^{n^2+2n}\sqrt{\sqrt{2n+2}+{\sqrt{n+1+\sqrt ...
16
votes
2answers
604 views

How to evaluate $\int_{0}^{1}{\frac{{{\ln }^{2}}\left( 1-x \right){{\ln }^{2}}\left( 1+x \right)}{1+x}dx}$

I want to evaluate $$\int_{0}^{1}{\frac{{{\ln }^{2}}\left( 1-x \right){{\ln }^{2}}\left( 1+x \right)}{1+x}dx}$$ I run this integral on Maple, It does converge. How we get a closed form? Is that ...
18
votes
2answers
460 views

How does one [easily] calculate $\sum\limits_{n=1}^\infty\frac{\mathrm{pop}(n)}{n(n+1)}$?

How does one [easily] calculate $\sum\limits_{n=1}^\infty\frac{\mathrm{pop}(n)}{n(n+1)}$, where $\mathrm{pop}(n)$ counts the number of bits '1' in the binary representation of $n$? Is there any trick ...
16
votes
1answer
286 views

Generalization of $\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor$

I have been asking the following question at MSE with an answer: $\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor$ is true? I found this relational ...
13
votes
2answers
336 views
+250

Is $\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}$ true for $m\in\mathbb N$?

Question : Is the following true for any $m\in\mathbb N$? $$\begin{align}\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}\qquad(\star)\end{align}$$ Motivation : I reached ...
21
votes
5answers
1k views

Sum of the form $r+r^2+r^4+\dots+r^{2^k} = \sum_{i=1}^k r^{2^k}$

I am wondering if there exists any formula for the following power series : $$S = r + r^2 + r^4 + r^8 + r^{16} + r^{32} + ...... + r^{2^k}$$ Is there any way to calculate the sum of above series (if ...
8
votes
4answers
413 views

Sum : $\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^3}$

Prove that : $$\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2k+1)^3}=\frac{\pi^3}{32}.$$ I think this is known (see here), I appreciate any hint or link for the solution (or the full solution).
25
votes
4answers
2k views

Identity involving Euler's totient function: $\sum \limits_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}$

Let $\varphi(n)$ be Euler's totient function, the number of positive integers less than or equal to $n$ and relatively prime to $n$. Challenge: Prove $$\sum_{k=1}^n \left\lfloor \frac{n}{k} ...
28
votes
1answer
612 views

Evaluating the sum $\lim_{n\to \infty}\sqrt[2]{2+\sqrt[3]{2+\sqrt[4]{2+\cdots+\sqrt[n]{2}}}}$

The following nested radical $$\lim_{n\to \infty}\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}$$ is known to converge to 2. We can consider a similar nested radical where the degree of the radicals increases: ...
7
votes
3answers
434 views

use residues to evaluate sum involving square of csch

I have been trying to evaluate the following sum using residues $\displaystyle \sum_{n=1}^{\infty}\frac{1}{\sinh^{2}(\pi n)}=\frac{1}{6}-\frac{1}{2\pi}$ I am mainly interested in using residues to ...
4
votes
1answer
105 views

Divisor function asymptotics

Define $\tau_{r}(n) = \sum_{d_1...d_r = n}1$. One exercise in a book on sieve theory asked for an elementary proof by induction of the fact that $$\sum_{n\le x}\tau_r(n) = \frac{1}{(r - 1)!}x(\ln ...