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

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0
votes
2answers
32 views

Ratio of 2 Sums of products of binomial coefficients

I want to prove that for $k \ even, 0 \leq k<n, n\in \mathbb{N}$: $-\frac{1}{(2n-3-k)(k+2)}\sum \limits_{i=0}^{k} \frac{(-1)^{i} 2^{i} (2n-2-i)!}{(n-1-i)!i!(k-i)!}=\sum \limits_{i=0}^{k+2} ...
1
vote
0answers
63 views

Summation verification

I have a particular polynomial $$ 1-10x+35x^2-50x^3 $$ Which can be written nicely as $$1-(1+2+3+4)x+(1\cdot2+1\cdot3+1\cdot4+2\cdot3+2\cdot4+3\cdot4)x^2$$ ...
1
vote
1answer
30 views

Does the following alternating series converge or diverge?

I have the following series that I have to check for convergence or divergence: $\sum\limits_{n=1}^{\infty} \frac{sin(n+ 1/2)\pi} {1 + \sqrt{n}}$ I know that it is an alternating series therefore I ...
2
votes
1answer
27 views

Swapping Two Sums

Suppose we have the following problem: $$\sum_{d=0}^{\infty} \sum_{s=0}^d f(s)g(d)$$ where $f$ and $g$ are two arbitrary function. If I want to sum over $s$ first and then $d$ what changes do I ...
-3
votes
1answer
77 views

Proving $\frac{200}{\pi}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)\cosh\left(\frac{\pi}{2}(2n+1)\right)}=25$

I solve a partial differential equation (Laplace equation) with specific boundary conditions and I finally found the answer: $$U(x,y)=\frac{400}{\pi}\sum_{n=0}^{\infty}\frac{\sin\left((2n+1)\pi ...
0
votes
1answer
45 views

How to solve the below scenario?

How can extract the individual no from its total sum for below case ? $$ 2^1+2^3+2^4+2^8+2^{13}+2^{17}=139546 $$ in the above expression i have only value 139546 but no knowledge about no of number ...
2
votes
0answers
40 views

How to prove that $\frac{200}{\pi}\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)\cosh{((2k+1)\frac{\pi}{2})}}=25$ [closed]

A friend of mine asked me to prove: $$ \frac{200}{\pi}\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)\cosh{((2k+1)\frac{\pi}{2})}}=25 $$ but I don't know where to start (honestly I am not much good at math). ...
0
votes
4answers
48 views

Convergent sequence? [duplicate]

Why does the $\lim \sum{n^{(1/n)}-1}$ diverge as $n\rightarrow \infty$? I suspected it would converge as $\lim {n^{(1/n)}}=1$ as $n\rightarrow \infty$ but computation show otherwise. So, now I am ...
0
votes
1answer
14 views

Iterated sums identity

How to show that the following iterated sums are equal? $\sum_i \sum_j f(i)h(j)g(i,j) = \sum_j\bigg(\sum_i g(i,j)f(i)\bigg) h(j)$
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votes
2answers
39 views

Factorial property of a double summation

I just proved the following identity, simply by expanding the sums: $$ \sum_{j=1}^h \sum_{i=0}^{j-1} f^i x_{j-i} = \sum_{j=1}^h \sum_{i=0}^{j-1} f^i x_{h-j+1} $$ I was wondering... is there a ...
0
votes
1answer
24 views

Number of prime divisors

Is there a way to express all the prime divisors of a natural number x as a function? Thanks in advance.
1
vote
1answer
25 views

Sum of all elements of a Set

Let's say I want to determine the number of natural numbers for an $x \in N$ this particular way: $$f(x) = \sum_{i=1}^x a\in\lbrace 1 : x\space\mathbf {mod}\space i=0\rbrace$$Is this the correct way ...
13
votes
11answers
3k views

Prove that 1 + 4 + 7 + · · · + 3n − 2 = n(3n − 1)/ 2

Prove that $$1 + 4 + 7 + · · · + 3n − 2 = \frac{n(3n − 1)} 2$$ for all positive integers $n$. Proof: $$1+4+7+\ldots +3(k+1)-2= \frac{(k + 1)[3(k+1)+1]}2$$ $$\frac{(k + 1)[3(k+1)+1]}2 + ...
2
votes
1answer
30 views

Exponential series is cosh(x), how to show using summation?

I want to show that $$\cosh(x) = \sum_{n=0}^{\infty} ‎\frac{(x)^{2n}‎}{(2n)!}‎ $$ I know that $cosh(x) = \frac{exp(x)+exp(-x)}{2}$ but i cant seem to get there from the original series. I know that ...
1
vote
3answers
18 views

Expression of Equation

Find the next three terms for the pattern following : (-1276),(+425),(-142),(+47) Simple pattern here.... Only pattern I can find is that, the odd number term goes up by 1134, and the even number ...
-1
votes
1answer
13 views

Does grouping the terms of a series (but not moving them) change the sum?

This question is an extension of this one, in which I am told that, given a sequence $a_1, a_2, a_3, ...$, $$\sum_{j=1}^{\infty }a_{j}=\sum_{n=1}^{\infty }(\sum_{k=2^{n-1}}^{2^{n}-1}a_{k})$$ is only ...
-1
votes
0answers
22 views

SUMMATION Proof via Induction [duplicate]

Provide a prove via induction that the SUM from i=1 to N of {i} is n(n+1)/2. Extend this proof to evaluate the SUM of i=1 to N of {i^k} for arbitrary natural number k.
0
votes
0answers
38 views

Summation of logarithms

I am trying to calculate the sum $\ln(a-x_1)+\ln(a-x_2)+....+\ln(a-x_n)$ and solve it somehow with respect to a ($x_1,x_2,....,x_n$ are measurements of a simulation) . The number of terms in the sum ...
2
votes
2answers
31 views

Summation notation for divided factorial.

I have the following sum $$5\cdot4\cdot3+5\cdot4\cdot2+5\cdot4\cdot1+5\cdot3\cdot2+5\cdot3\cdot1+$$$$5\cdot2\cdot1+4\cdot3\cdot2+4\cdot3\cdot1+4\cdot2\cdot1+3\cdot2\cdot1$$ It is basically $5!$ ...
1
vote
1answer
42 views

When does equality hold in $\bigg(\sum_{k=1}^na_k^2\bigg)\bigg(\sum_{k=1}^nb_k^2\bigg)\ge\bigg(\sum_{k=1}^na_kb_k\bigg)^2$

In the Cauchy-Schwarz inequality:$$\bigg(\sum_{k=1}^na_k^2\bigg)\bigg(\sum_{k=1}^nb_k^2\bigg)\ge\bigg(\sum_{k=1}^na_kb_k\bigg)^2$$If $a=<a_1,a_2,...,a_n>$ and $b=<b_1,b_2,...,b_n>$ then ...
0
votes
3answers
61 views

What is the substitution to remove index variable from inside a sum? [closed]

For instance, how to remove the m from inside this sum? $$\sum_{r=0}^{m-1} f\left(\frac rm\right)$$ or $$\sum_{r=0}^{m} f\left(\frac rm\right)$$
0
votes
1answer
42 views

Inverse of $f(x) = \sum_{i=0}^x \lfloor 2 \pi i \rfloor$

Is there a closed expression that could be used to compute the inverse of $f(x) = \sum_{i=0}^x \lfloor 2 \pi i \rfloor$? Or at least an algorithm with low computational complexity for high x?
10
votes
2answers
71 views

Calculate the sum : $S=1+\frac{\sin x}{\sin x}+\frac{\sin2x}{\sin^2 x}+\cdots +\frac{\sin nx}{\sin^n x}$

I was given a task to calculate this sum: $$S=1+\frac{\sin x}{\sin x}+\frac{\sin 2x}{\sin^2 x}+\cdots +\frac{\sin nx}{\sin^n x}$$ but I'm not really sure how to start solving it. Like always, I would ...
1
vote
2answers
113 views

Compute the following sum $ \sum_{i=0}^{n} \binom{n}{i}(i+1)^{i-1}(n - i + 1) ^ {n - i - 1}$?

I have the sum $$ \sum_{i=0}^{n} \binom{n}{i}\cdot (i+1)^{i-1}\cdot(n - i + 1) ^ {n - i - 1},$$ but I don't know how to compute it. It's not for a homework, it's for a graph theory problem that I try ...
0
votes
1answer
45 views

to find total number of subsets

I was working out some problem where I needed permutation and combination. I took the cartesian product of $n$ sets where number of elements in each set is even and $n$ is odd. Further the elements of ...
0
votes
4answers
50 views

What is the sum of infinte values of x, where x tends to zero? Is the sum 0, or does it go on to become infinite?

I feel that the sum of all very small numbers should be ultimately infinite, however small they are. But I also feel that since all are tending to zero, even the sum will tend to zero? Which is right, ...
1
vote
1answer
27 views

What is the substitution to remove integration limit from inside an integral?

For instance, how to remove the $m$ from inside this integral? $$\int_0^m f\left(\frac rm\right) dr$$
-3
votes
2answers
41 views

Use mathematical induction to prove $\sum_{i=1}^{n}(2i+4)=n^2+5n$

Prove: $$ \sum_{i=1}^{n}(2i+4)=n^2+5n \textrm{ for each positive integer } n $$ So I'm not exactly sure how to do this problem for my math class. Can any mathematicians out there help me? ...
0
votes
2answers
42 views

For which of the following choice of $a_k$ is $\sum a_k$ convergent?

For which of the following choice of $a_k$ is $\sum a_k$ convergent? i)$\frac {sinh(k)}{2^k}$ ii)$(1-\frac{1}{k})^{k^2}$ Honestly, I have no idea. Usually, when I see $sin$ or $cos$, I use ...
2
votes
1answer
77 views

Prove $\sum_{k=1}^{\infty} \frac{\sin(kx)}{k} $ converges

How to prove $$\sum_{k=1}^{\infty} \frac{\sin(kx)}{k}$$ converges without using integral test?
1
vote
2answers
33 views

Do these converge?

For the following choices for $a_k$, use the indicated test to show whether $\sum a_k$ converges or diverges $\frac{1}{k^{1/k}k}$ (Comparison Test, limit form) $\binom{2k}{k}^{-1}(4-10^{-23})^k$ ...
2
votes
1answer
67 views

Bounds on sum of cosines

Find bound on the following sums for $k\in \{1,\dots p-1\}$ where $p$ is a prime (even assumed to be $3\mod 4$, I used this to get to the current sum) find good upper and lower bounds on ...
1
vote
2answers
60 views

By expanding $e^x$ into a series prove the following inequality

By expanding $e^x$ into a series $\sum e^x$ prove that $$\forall x \in \mathbb{R}, x \ge 0 \implies e^{x-1} \ge x$$ Also show when this inequality becomes equality. I'm not really sure how to attack ...
3
votes
5answers
141 views

How do I succinctly note the sum of $(n-1)+(n-2)+…$?

I was playing with numbers and wanted to see how many possible connections there are in a network of $n$ nodes. I found that the answer was equal to ...
2
votes
1answer
43 views

Absolute Value Trig Sum

I have been trying to solve $$y(x)=\sum_{k=1}^{\infty} \frac{|\cos(kx)|}{k}$$ however, this is proving to be more difficult than I had hoped, and cannot seem to figure this out. What I have figured ...
0
votes
2answers
30 views

Prove convergence by considering the partial sums

Let $p$ be a non-zero natural number. Prove by considering the partial sums that $\sum \frac{1}{k(k+p)}$ converges. What is $\sum\limits_{k=1}^{\infty} \frac{1}{k(k+p)}$ No idea. Obviously, it ...
0
votes
1answer
22 views

Algebra question $n\sum_{k=0}^n {n-1 \choose k-1} p^{k}q^{n-k} = np\sum_{k=1}^n {n-1 \choose k-1} p^{k-1}q^{n-k}$

In a proof in my textbook one step goes from ... $$n\sum_{k=0}^n {n-1 \choose k-1} p^{k}q^{n-k} = np\sum_{k=1}^n {n-1 \choose k-1} p^{k-1}q^{n-k}$$ I understand that you can take the $p$ out because ...
12
votes
4answers
275 views

Which expansion of $e$ is more accurate?

We have two forms of $e^x$ $$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+....$$ and $$e^x=\frac{1}{\displaystyle 1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+....}$$ The second form comes from ...
0
votes
2answers
70 views

If $\sum a_n$ converges, does $\sum a_n / 2^n$ converge as well?

If $\sum a_n$ converges, does $\sum \dfrac{a_n}{2^n}$ converge as well? I can't use differential or integral calculus for this.
-1
votes
1answer
63 views

Does the series $\sum \sin(100n)$ converge? [duplicate]

Does the following series converge? $$\sum \sin(100n) = \sin(100) + \sin(200) + \dots$$
2
votes
0answers
34 views

How the second form of following equation is derived form first form (i.e. given first line, what are the steps involved in writing second line

How the second form of following equation is derived form first form (i.e. what are the steps involved in writing second line)
1
vote
2answers
53 views

Is there any way to approximate a sum of square roots

I am trying to calculate a sum of square roots $\sum\limits_{i=1}^n \sqrt{a + i}$ and after some struggling and googling I gave up on this. Is there any way to get a closed formula for this sum ...
3
votes
2answers
38 views

Summation of trigonometric functions such as $\sin x$

I am currently studying Integration (a very basic introduction) and I have a question regarding the summation of trigonometric functions. Given $f(x) = \sin x$, determine the area under the curve ...
1
vote
5answers
59 views

The limit of a sum of the form $a_0 \sqrt n + a_1 \sqrt {n + 1} +\cdots + a_k \sqrt {n + k}$

If $ a_0 ,a_1 ,\ldots,a_k $ are real numbers such that $$ a_0 + a_1 + \cdots + a_k = 0$$ Find $$ \lim_{n \to \infty } (a_0 \sqrt n + a_1 \sqrt {n + 1} +\cdots + a_k \sqrt {n + k} )$$ I just ...
18
votes
1answer
600 views

How to prove a double sum is always an integer?

I have verified the following double sum is always an integer for $s$ up to $1000$ via Maple. But I can not prove it. Proofs, hints, or references are all welcome. Thanks! ...
1
vote
0answers
32 views

How should i go about proving an expression of this kind?

Lets say i have a complete bell polynomial that is equal to a summation such that $$ B_n(d_1,d_2,\cdots,d_n) = \sum_{k=0}^{n}[g(x)^{-k} h(k)] $$ Where $d_n = \frac{d^n}{dx^n}[f(x)\ln(g(x)]$ and ...
3
votes
1answer
45 views

Prove difference of summations $=\frac{e^2}{2}$

How do I prove that \begin{align} ...
1
vote
1answer
26 views

What is $s_3$ and $s_4$ for $x$

$\sum_{i=0}^n i^k = s_k(n)$, $s_k$ polynomial from degree $k+1$ I have already shown for $s_2(x) = \frac{x(x+1)(2x+1)}6$ How from the sum and $s_2(x)$ can be shown for $s_3(x)$ and $s_4(x)$ ...
0
votes
2answers
34 views

What is the value of the following summation?

Compute $$\displaystyle\sum \limits_{n=0}^\infty (-1)^{n+1} \frac{1}{9^n(2n+2)}$$ I am given the fact that $$ \frac{1}{2}\ln(1+x^2) = \sum \limits_{n=0}^\infty (-1)^n\frac{x^{2n+2}}{2n+2} $$ ...
0
votes
2answers
74 views

Using mathematical induction to prove $\frac{1}1+\frac{1}4+\frac{1}9+\cdots+\frac{1}{n^2}<\frac{4n}{2n+1}$

This induction problem is giving me a pretty hard time: $$\frac{1}1+\frac{1}4+\frac{1}9+\cdots+\frac{1}{n^2}<\frac{4n}{2n+1}$$ I am struggling because my math teacher explained us that in ...