Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.
8
votes
4answers
278 views
Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction
How can I prove that
$$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$
for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction.
Thanks
14
votes
1answer
436 views
A nice log trig integral
Show that :
$$\int_{0}^{\frac{\pi }{2}}{\frac{{{\ln }^{2}}\cos x{{\ln }^{2}}\sin x}{\cos x\sin x}}\text{d}x=\frac{1}{4}\left( 2\zeta \left( 5 \right)-\zeta \left( 2 \right)\zeta \left( 3 \right) ...
2
votes
5answers
123 views
Proof via Induction for A Summation
I'm starting to understand how induction works (with the whole $k \to k+1$ thing), but I'm not exactly sure how summations play a role. I'm a bit confused by this question specifically:
$$
...
6
votes
6answers
778 views
Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction
I am just starting out learning mathematical induction and I got this homework question to prove with induction but I am not managing.
$$\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$$
...
5
votes
2answers
101 views
Sum up to number $N$ using $1,2$ and $3$
So the question asked was finding out the number of ways(combinations), a given number $N$ can be formed using the sum of $1,2$ or $3$.
(eg)
For $n = 8$, the answer is $10$
The given solution for ...
4
votes
1answer
132 views
Sum involving the hypergeometric function, power and factorial functions
I am finding some trouble in calculating the following sum involving the hypergeometric function, power and factorial functions.
$$
\sum_{y=1}^\infty \mathrm{e}^z \cdot {}_1F_1\left(1-y;2;-z\right) ...
4
votes
2answers
263 views
Inductive proof that ${2n\choose n}=\sum{n\choose i}^2.$
I would like to prove inductively that $${2n\choose n}=\sum_{i=0}^n{n\choose i}^2.$$
I know a couple of non-inductive proofs, but I can't do it this way. The inductive step eludes me. I tried naively ...
1
vote
2answers
41 views
Summations with binomial coefficients:$\sum_{k=0}^{n}\binom{R}{k}\binom{M}{n-k}=\binom{R+M}{n}$
Can someone help me solve this equation? How to prove that $$\sum_{k=0}^{n}\binom{R}{k}\binom{M}{n-k}=\binom{R+M}{n}?$$
1
vote
4answers
118 views
Computing $\sum_{i=1}^{n}i^{k}(n+1-i)$
I dont know how to proceed with solving $$\sum_{i=1}^{n}i^{k}(n+1-i).$$ Please give advise.
0
votes
1answer
106 views
Expected value of a Poisson sum of confluent hypergeometric functions
How to compute the expected value of a Poisson sum of the following confluent hypergeometric function:
$$
\sum_{y=1}^{Y} {}_1F_1(y,1,z)
$$
where y is positive integer taking values from the Poisson ...
16
votes
0answers
658 views
Prove that sum is finite
Let $j \in \mathbb{N}$. Set
$$
a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!}
$$
and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$.
Please help me to prove that the following sum is ...
8
votes
1answer
147 views
Prove that $\sum_{k=0}^n{e^{ik^2}} = o(n^\alpha)$, $ \forall \alpha >0$
I want to prove that :
$\sum_{k=0}^n{e^{ik^2}} = o(n^\alpha)$, $ \forall \alpha >0$ when $n$ tends to $+\infty$
Perhaps $\sum_{k=0}^n{e^{ik^2}}$ is bounded, I don't know.
Do you have ideas ?
10
votes
1answer
359 views
How to evaluate $\int_{0}^{1}{\frac{{{\ln }^{2}}\left( 1-x \right){{\ln }^{2}}\left( 1+x \right)}{1+x}dx}$
I want to evaluate $$\int_{0}^{1}{\frac{{{\ln }^{2}}\left( 1-x \right){{\ln }^{2}}\left( 1+x \right)}{1+x}dx}$$
I run this integral on Maple, It does converge. How we get a closed form?
Is that ...
8
votes
4answers
307 views
Sum : $\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^3}$
Prove that : $$\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2k+1)^3}=\frac{\pi^3}{32}.$$
I think this is known (see here), I appreciate any hint or link for the solution (or the full solution).
6
votes
1answer
72 views
Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^a H_n}{2^n}$
Is there a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^a H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$
This is a ...
9
votes
3answers
107 views
Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n}$
Please help me to find a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$
8
votes
3answers
226 views
sum of a series
Can
\begin{equation}
\sum_{k\geq 0}\frac{\left( -1\right) ^{k}\left( 2k+1\right) }{\left(
2k+1\right) ^{2}+a^{2}},
\end{equation}
be summed explicitly, where $a$ is a constant real number? If $a=0,$ ...
9
votes
2answers
391 views
Does this double series converge?
$$\sum\limits_{y=1}^{Y}\sum\limits_{z=1}^{y} a^{y-1} b^y \binom{y-1}{z-1} (c + 2z)^d $$
Does this series converge when $Y=∞$? If the series converges, what does it converge to? If the series does not ...
5
votes
2answers
57 views
Binomial probability with summation
Show that
$$\sum_{k=0}^{m} \frac{m!(n-k)!}{n!(m-k)!} = \frac{n+1}{n-m+1}$$
Attempt:
It becomes:
$$\sum_{k=0}^{m } \frac{\binom{m}{k}}{\binom{n}{k}}$$
Telescoping, pairing, binomial theorem don't ...
1
vote
1answer
92 views
How to calculate $\sum_{k=1}^{k=n}\frac{\sin(kx)}{\sin^{k}(x)}$?
I was given an exercise:
Calculate 1+$\sum_{k=1}^{k=n}\frac{\sin(kx)}{\sin^{k}(x)}$
I recognize $$\sin(kx)=Im(cis(kx))=Im(cis^{k}(x))$$ and $$\sin^{k}(x)=(Im(cis(x)))^{k}$$
but I do not know ...
8
votes
3answers
100 views
Help me prove this inequality :
How would I go about proving this?
$$ \displaystyle\sum_{r=1}^{n} \left( 1 + \dfrac{1}{2r} \right)^{2r} \leq n \displaystyle\sum_{r=0}^{n+1} \displaystyle\binom{n+1}{r} \left( \dfrac{1}{n+1} ...
7
votes
6answers
263 views
simplify summation of factorial (random walk)
I suspect that the expression
$$\sum_{n=0}^N \frac{(N-2n)^2}{n!(N-n)!}$$
simplifies to
$$\frac{2^N}{(N-1)!}$$
But I cannot find the intermediate steps. Can someone give me a hint how I can deduce ...
5
votes
2answers
139 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 ...
3
votes
2answers
233 views
Partial sum of an alternating series: $2 - \frac{4}{3} + \frac{8}{9} - \cdots + \frac{(-1)^{20}2^{21}}{3^{20}}$
I know the sum of the series $$2 - \frac{4}{3} + \frac{8}{9} - \cdots + \frac{(-1)^{20}2^{21}}{3^{20}}$$ is equal to $$\sum\limits_{n=0}^{20} \frac{(-1)^{n}2^{n+1}}{3^{n}},$$ but I don't know how to ...
2
votes
3answers
150 views
When is a factorial of a number equal to its triangular number?
Consider the set of all natural numbers $n$ for which the following proposition is true.
$$\sum_{k=1}^{n} k = \prod_{k=1}^{n} k$$
Here's an example:
$$\sum_{k=1}^{3}k = 1+2+3 = 6 = 1\cdot 2\cdot ...
1
vote
1answer
108 views
$ \sum_{y=1}^\infty {}_1F_1(1-y;2;-\pi\lambda c) \frac{\lambda^y}{y!} $
I am not able to solve the following sum. Can you please provide any hints ?
$$ \sum_{y=1}^\infty {}_1F_1(1-y;2;-\pi\lambda c) \frac{\lambda^y}{y!} $$
Note that the 3rd parameter of the Confluent ...
1
vote
0answers
77 views
Summation of the series $s_b(p)=\sum_k b^{k^p}$ by a double sum in a sense like Ramanujan-method
From some older context I am re-considering the following variant of the geometric series
$$ s_b(p)=\sum_{k=1}^\infty b^{k^p} $$
for the convergent cases $0 \lt b \lt 1$ and $ 0 \lt p$ first. I'm ...
6
votes
3answers
162 views
What is the probability that $n$ dice tie on successive rolls?
The Question
What is the probability, rolling $n$ six-sided dice twice, that their sum each time totals to the same amount? For example, if $n = 4$, and we roll $1,3,4,6$ and $2,2,5,5$, adding them ...
5
votes
1answer
81 views
$\sum_{i=0}^m \binom{m-i}{j}\binom{n+i}{k} =\binom{m + n + 1}{j+k+1}$ Combinatorial proof
Is there a simple combinatorial proof for the following identity?
$$\sum_{0\leq i \leq m} \binom{m-i}{j}\binom{n+i}{k} =\binom{m + n + 1}{j+k+1}$$
where $m,j \geq 0$, $k \geq n \geq 0$.
5
votes
3answers
522 views
Complete induction
I am very confused with complete induction. Because in every task there is something different to do, and I never know what to insert (thats my biggest problem).
Here's the example:
Proof with ...
4
votes
1answer
184 views
sum of $\displaystyle \frac{\sin nx }{n^4}$
Consider : $\displaystyle f(x)= \sum_{n=1}^{\infty} \frac{\sin nx }{n^4}$
Find : $\displaystyle \int_0^{x} f(t)\ \mathrm{d}t$.
4
votes
4answers
118 views
Binomial series with two binomial coefficents
My question reads: Does this formula has mathematical meaning at first place? Is it summable?
$$\sum^{\infty}_{k=0}{n\choose k}{m\choose k} x^k$$
3
votes
1answer
58 views
Show that $\frac{1}{1+x}H(\frac{x}{1+x})=\sum^\infty_{k=0}[\Delta^kh_0]x^k$
For a sequence $\{h_n\}_{\geq 0}$, let $H(x)=\sum_{n\geq0}h_nx^n$. Show that:
$$\frac{1}{1+x}H(\frac{x}{1+x})=\sum^\infty_{k=0}[\Delta^kh_0]x^k$$
What I did was that by proving the $$\Delta^k ...
3
votes
2answers
168 views
Closed form sum of $\sum^{\infty}_{n=1} \frac{1}{3^n-1}$
Wolframalpha uses $q$-Polygamma function to represent the sum, hence essentially does nothing. Here I wonder if this sum can be represented by elementary function.
The summation is like a infinite ...
2
votes
0answers
38 views
Sum of squared/cube combinations [duplicate]
I was wondering if there is a closed formula for sum of cubed combinations. More precisely, I'd like to compute $$\sum_{k=1}^n \left ( \begin{array}{c}n\\k\end{array}\right )^3$$
Obviously, without ...
2
votes
1answer
48 views
Sum of squared/cubed combinations
I was wondering if there is a closed formula for sum of cubed combinations. More precisely, I'd like to compute $$\sum_{k=1}^n \left ( \begin{array}{c}n\\k\end{array}\right )^3$$
Obviously, without ...
2
votes
1answer
53 views
Expected number of edges: does $\sum\limits_{k=1}^m k \binom{m}{k} p^k (1-p)^{m-k} = mp$
Find the expected number of edges in $G \in \mathcal G(n,p)$.
Method $1$: Let $\binom{n}{2} = m$. The probability that any set of edges $|X| = k$ is the set of edges in $G$ is $p^k (1-p)^{m-k}$. ...
2
votes
0answers
76 views
Proof for a summation-procedure using the matrix of Eulerian numbers?
I've discussed a procedure for divergent summation using the matrix of Eulerian numbers occasionally in the last years (initially here, and here in MSE and MO but not in that generality and thus(?) ...
2
votes
1answer
85 views
How to simpify the following equation involving binomial coefficients?
How can one simplify this equation:
$$
\sum_{k=0}^{n-1}\binom{n}{k}\binom{n}{k+1}
$$
2
votes
2answers
67 views
Summing $r(r+3)$ using induction
We want to prove the following summation by induction:
$$\sum_{r=1}^{n}r(r+3)=\frac{1}{3}n(n+1)(n+5)$$
The problem is posted for a friend, but others can look at the solution if they want/need.
2
votes
3answers
116 views
How to solve second degree recurrence relation?
For first degree recurence relation it is as simple as $f(n)=a^n\cdot f(0)+b\dfrac{a^n-1}{a-1}$.
But how do you solve second degree?
For example
$$f(n)=\begin{cases}
1,&\text{for }n=1\\
...
1
vote
1answer
42 views
$\sum_{i=1}^{n} (3i + 2n)$
I want to verify what would be the simplified solved version of this summation.
$$\sum_{i=1}^{n} (3i + 2n)$$
Would it be this?
$$ \frac32n^2 + \frac32n + 2n^2 $$
1
vote
1answer
104 views
Multiplying two series together
How would I multiply two series together? Or also split them into two separate series? For example:
$$\sum_{y=1}^{b}\sum_{x=1}^{a}2^{(2x+3y)}$$
I tried multiplying the summation of $4^x$ with ...
1
vote
3answers
108 views
Can the identity $n(n+1)2^{n-2} = \sum_{i=1}^{n} i^2 \binom{n}{i}$ be derived from the binomial theorem?
Can this identity be derived from the binomial theorem?
$n(n+1)2^{n-2} = \sum_{i=1}^{n} i^2 \binom{n}{i}$
Please, explain how.
I tried starting from $2^n = \sum_{i=0}^{n} \binom{n}{i}$ and ...
1
vote
3answers
137 views
Finding summation
I have been having problem with calculating the following summation:
$$
\sum_{i=1}^n {1\over 4i^2-1} = {1\over3} + {1\over15} + {1\over35} + \cdots + {1\over 4n^2-1}
$$
I do know the answer, but just ...
1
vote
4answers
137 views
Find $\sum_{n=1}^{\infty}\frac{1}{n!}$
Find $$\sum_{n=1}^{\infty}\frac{1}{n!}$$
All of the advice I've seen to compute this sum says to use the ratio test, but this is in a chapter BEFORE the ratio test, so the book wants me to solve ...
1
vote
1answer
149 views
How can the Möbius function be applied to a series?
Given a series $p_n(s)=\sum_{k=1}^n a_k $. I'd like to get $\hat{p}_n(s)=\sum_{k=1}^n \mu(k)a_k $. Think of $a_k=k^{-s}$ for example. If you let $n$ go to $\infty$, you'll see the well-known relation ...
0
votes
0answers
56 views
Expected value of a Poisson sum of confluent hypergeometric functions times (version 2)
In continuation to my question here , what is the expected value of a Poisson sum of the following confluent hypergeometric function:
$$
\sum_{y=1}^{Y} (1/Y)({}_1F_1(y,1,z))
$$
where y is positive ...
0
votes
1answer
48 views
Manipulating double summation
In a problem in my book, the following equality is there:
$$\sum_{n=0}^\infty\Big( \sum_{k_i\ge 0, \sum_{i=1}^\infty ik_i=n}\frac{x^n}{\prod_{i=1}^\infty k_i!(i!)^{k_i}}\Big)=\sum_{k_i\ge ...
0
votes
2answers
420 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| ...
