# Questions tagged [summation]

11,411 questions
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### What is the limit of $\sum_{k=1}^{n}\frac{k^3}{n^4}$? [closed]

Find the following limit: $$\lim_{n \to \infty} \sum_{k=1}^{n}\frac{k^3}{n^4}$$ Should I approximate this with integrals?
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### $\sum\limits_{n=1}^{\infty}\sin ( \frac{n}{2^n})$ converges? [closed]

I was trying to determine weather or not $\sum\limits_{n=1}^{\infty}\sin ( \frac{n}{2^n})$ converges using perhaps the D'Alembert test, but given the sine I cant really see it happening..are there ...
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### Induction proof of $1+3+\cdots+3^n=\frac{3^{n+1}-1}{2}$ [closed]

How would I prove the following by induction?$$1+3+3^2+3^3+\cdots+3^n=\frac{3^{n+1}-1}{2}$$ for all $n\geq 0.$ I kept trying to create a base case but I am not sure how many I need. I also seem to be ...
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### How can we prove that $27$ is the largest number $χ$ such that $\sum_{i \in δ(χ^3)} = χ$? [closed]

According to Stetson University, $27$ is the largest number $χ$ such that $\sum_{i \in δ(χ^3)} = χ$ where $δ(χ) =$ digits of $χ$. I have tried proving that this is true, but do not know where to ...
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### Sum of numerators divided by sum of denominators $\leq$ the maximum fraction [duplicate]

Let $\tfrac{a_1}{b_1},\dots,\tfrac{a_n}{b_n}$ where $a_i,b_i>0$. How can one prove that $$\frac{\sum_i a_i}{\sum_i b_i}\leq \max_j \tfrac{a_j}{b_j}$$?
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### $\sum_{n=1}^\infty x(1-x)^{n-1}$ Does this sum converge uniformly?

$$\sum_{n=1}^\infty x(1-x)^{n-1}$$ I know that this sum converge $\iff$ $0\le x \le 1$, i wanted to use the Weierstrass but could not suceed, so i think this sum might not converge uniformly,but i'm ...
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### Summing this series??

Let $T_n$ = $\sqrt{2000^2 - n^2}$ The sum of this series to n terms is $S_n$ Then find $\frac{2}{2000^2}(2000 + 2S_{2000})$ upto $2$ decimal places. I know in homework questions we do have to ...
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### Remainder of $\sum_{x=1}^{312} x \times x!$ divided by 2016 [closed]

I have this question: What is the remainder of $$\sum_{x=1}^{312} x \times x!$$ (or just simply) $$(1! \times 1) + (2! \times 2) + (3! \times 3) + \dots + (312! \times312)$$ Divided by ...
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### A Riemann-type sum [closed]

I want to solve this summation, however I have no idea where to start. Could any one help me find a good starting place? $$\sum_{i=1}^{n}\sin\left(i \over n\right)\frac{1}{n}$$ Any help would be ...
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### $\sum_{i=1}^n \frac{n}{\text{gcd}(i,n)}.$ [closed]

Find the value of this series: $$\sum_{i=1}^n \frac{n}{\text{gcd}(i,n)}.$$
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### Show that $\sum \limits_{i=1}^n log(i)$ is O(n log n) [duplicate]

How can I show that $\sum \limits_{i=1}^n \log(i)$ is $O(n \log n)?$ (Log in base 2).
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### What is the sum $\sum_{k=10}^{\infty}\left(\frac{1}{2x}\right)^k$ [closed]

I would appreciate some directions regarding the follow problem, $\sum_{k=10}^{\infty}\left(\frac{1}{2x}\right)^k=$?
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Statement:If $a_1,a_2,a_3\cdots a_n$ be $n$ unequal and positive quantities and if $m>r>0$ , then $$\frac{a_1^{m}+a_2^{m}\cdots +a_n^{m}}{n}> \frac{a_1^{r}+a_2^{r}\cdots +a_n^{r}}{n}. \frac{... 1answer 68 views ### Question concerning sigma notation [duplicate] Consider you have been given that$$\sum_{i = 1}^{\infty}i = -\dfrac{1}{12} $$How do you solve this sigma notation? I've not seen this kinda sigma notation before. Regards! 1answer 53 views ### Please simplify this sigma question? [closed] I am not able to solve this sigma question. Please anybody solve this question by steps.$$\sum_{R=1}^N\left(\frac13\right)^{R-1}$$3answers 407 views ### Series into sigma notation How do I convert -1 + 3 - 5 +...- 101 into sigma notation. I tried to divide the series into -1 -5 -7 -... and 3+7+9 but i'm not too sure if that is correct. 2answers 51 views ### Evaluating the following sums with these suppositions. I really need help with this. [closed] I have tried everything and I am just unable to solve the following sums. Mainly because I do not understand the suppositions and why they are there in the first place and further I do not get what ... 2answers 57 views ### Find the partial sum of S_n [closed] EDİT: How we can calculate S_n for any k with MATRİX METHOD? S_n=1^k+2^k+3^k+...+n^k k\in$$\mathbb{N}$1answer 38 views ### Solve$\sum_{i=1}^{200} {1\over{1+x_i}} =?$[closed] $$(x^{2}+x+1)^{100}=a_0+a_1x+a_2x^{2}+...+a_{199}x^{199}+a_{200}x^{200}$$ $$\sum_{i=1}^{200} {1\over{1+x_i}} =?$$ Can somebody help me? Thank you! 1answer 832 views ### what is the summation from i=0 to log(n) [closed] I need to know how to get the summation of a constant (c) from i=0 to log(n) of a constant 1answer 186 views ###$((n-1)^{0.5})/(((n+1)^2)-1)$Is the sum convergent?, why or why not? [closed] $$\frac{(n-1)^{0.5}}{(n+1)^2-1}$$ Sorry I dont know how to to do sub or superscripts. I would like a step by step method please, thanks. 2answers 46 views ### Prove by induction that$\sum_{i=1}^n i \geq \frac{n^2}{2}\$ [closed]

Can someone show me a formal proof of this exercise ? $$\sum\limits_{i=1}^n i \geq \frac{n^2}{2}, \quad \forall n \in \mathbb{N}.$$ Thanks to anyone who can help! :)
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### Sum of finite series involving square roots [duplicate]

What's the the result of: $$\sum_{k=1}^{n}{\sqrt{k}+1}$$ Thanks.
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### Summation Problems [closed]

How did this particular equation come about? I haven't seen it before in the summation rules index on wikipedia: $$\sum\limits_{i=1}^{k+1} x_i =\left(\sum\limits_{i=1}^{k} x_i\right)+x_{k+1}$$ Edit:...
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