Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [summation]

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

The notation $\sum\limits_{i=1}^na_i$ means $a_1+\ldots +a_n$.

Use for sums of infinite series and questions of convergence; use for questions about finite sums and simplification of expressions involving sums.

17770 questions
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Formula for $\sum _{i=1}^n (n+1-i) (n-i)$

It is easy to show that $$\sum _{i=1}^n (n+1-i) (n-i) = n (n-1)+(n-1) (n-2)+...+1 (1-1)=\frac{1}{3} \left(n^3-n\right)$$ using induction. But how do I derive this formula? I couldn't find any substitution to do this.
Max
  • 125
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3 answers

Sum of certain consecutive numbers is $1000$.

Question: The sum of a certain number (say $n$) of consecutive positive integers is $1000$. Find these integers. I have no idea how to approach the problem. I did try the following but did not arrive anywhere: I said…
AryanSonwatikar
  • 2,728
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3 answers

How do you compute the sum of k * a^k

We have the sum $$\sum_{k=0}^{n} a^k k,$$ where a is a constant and we need the answer in terms of $n$. How can we go about solving this? If $a$ were a variable we could use differentiation with $\sum_{k=0}^{n} a^k$, but I don't think we can…
Pro Q
  • 915
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Why is $\sum_{r=1}^n r^{\overline{a+1}}(r+b)=\big(\sum_{r=1}^n r^{\overline{a+1}}\big)\big(An+B\big)$?

Why is $$\sum_{r=1}^n r^{\overline{a+1}}(r+b)=\bigg(\sum_{r=1}^n r^{\overline{a+1}}\bigg)\bigg(An+B\bigg)=\frac{n^{\overline{a+2}}}{a+2}\cdot (An+B)$$ where $A, B$ are rational numbers? ? It is clear that LHS is a polynomial in $n$ of order…
Hypergeometricx
  • 22,657
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Prove $\sum_{i=1}^n a_i$ = $\sum_{i=2}^{n+1} a_{i-1}$

Given $\sum_{i=1}^n a_i$ = $\sum_{i=2}^{n+1} a_{i-1}$ How would you show this true for all n ∈ N and $a_1, a_2, . . . , a_n$ ∈ R? I know it is obviously true because i would just use a substitution like i=j-1 then summing j-1 from 2 to n+1 gives the…
Harry
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Relabelling indices in multiple summations

Suppose I have a summation that looks like \begin{align} \sum_{a=0}^{n/2} \sum_{b=0}^a \sum_{c=0}^{n-2b}\sum_{d=0}^c \alpha(a,b,c,d)f(c-2d), \end{align} where $\alpha$ and $f$ are functions of the indices $a,b,c,d$, and I want to change these…
K L
  • 123
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Is this correct? Convergence of $\sum_{n=2}^\infty\frac1{\log\left(\frac{n(n+1)!}2\right)}$

Is this correct? $$\sum_{n=2}^\infty\frac1{\log\left(\frac{n(n+1)!}2\right)}<\sum_{n=2}^\infty\frac{1}{\log n!}\approx\sum_{n=2}^\infty\frac1{n\log (n)-n}$$ Since the integral below does not converge, then the sum does not also…
Euleroid
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Advice on Evaluating $\frac{\sum _{n=1}^m(n \times\log_2(n))}{m \times\log_2(m)}$

I recently came across this question: Evaluate: $$\frac{\sum _{n=1}^m(n \times\log_2(n))}{m \times\log_2(m)}$$ However, I'm not certain where to begin, I considered finding the bounds to the equation by integration, but I don't know if there's any…
DoublyNegative
  • 185
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1 answer

When can I assume two different summations count at the same time?

Suppose I have $$ \sum_{n=1}^{m}n \ \ + \ \ \sum_{k=1}^{m}k.$$ This isn't a generalized case, but I'm not sure what the extent of the generalized case is to begin with. Can I assume that $n=k$ and have both summations count to $m$ at simultaneously?…
user19788
  • 29
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Finding $\sum_{a_1=2}^{9}{...\sum_{a_2=a_1}^{9}{\sum_{a_n=a_{n-1}}^{9}{a_n}}}$

Let $$f(n)=\sum_{a_1=2}^{9}{\sum_{a_2=a_1}^{9}{\sum_{a_3=a_2}^{9}{...\sum_{a_n=a_{n-1}}^{9}{a_n}}}}$$ A) How could one find $$\sum_{k=1}^{n}{f(k)}?$$ B) How could one find how many terms there are in the sum? For Part B, I know that the number of…
Shrey Joshi
  • 596
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2 answers

Help to understand $\sum_{1 \leqslant k \leqslant n} \sum_{1 \leqslant j \lt k} {\frac{1}{k-j}}$

I got this summation from the book Concrete Mathematics which I didn't exactly understand: $$ \begin{align} Sn &= \sum_{1 \leqslant k \leqslant n} \sum_{1 \leqslant j \lt k} {\frac{1}{k-j}} \\ &= \sum_{1 \leqslant k \leqslant n} \sum_{1 \leqslant…
Diego Marques
  • 91
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2 answers

Finite double summation - the upper limit of the inner summation depends on the outer summation's parameter

I am doing some work on finding the distribution of a sum of two independent random variables. In my actual work these two variables are independent, yet have a very different distribution. I wanted to treat an example case, but I lack the…
user60307
2
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2 answers

Finding a general formula for summation

I am trying to find a general summation for $$\sum_{k=1}^{n} \frac{(k-1)2^k}{k(k+1)}$$ I tried to expand it and look for some pattern. $$\frac{(1-1)2^1}{1(1+1)} + \frac{(2-1)2^2}{2(2+1)}+...+\frac{((k-1)-1)2^{k-1}}{(k-1)((k-1)+1} +…
AdamK
  • 333
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4 answers

Simplify a summation with indicator function

I have this double summation that I want to simplify: $$\sum^{q} _{j=-q} \sum^{q} _{i=-q} \mathbb{1}_{ \{h+j-i=0 \}}$$ A given solution says the answer is $2q+1-h$ for $|h| \leq 2q$ and $0$ otherwise. I don't see this. For example, when I take…
clubkli
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