All Questions
31
questions
3
votes
4
answers
98
views
Compute $\sum_{k=1}^{25} (\frac{1}{k}-\frac{1}{k+4})$
Compute $\sum_{k=1}^{25} (\frac{1}{k}-\frac{1}{k+4})$
I know that some of the terms will cancel each other. Have it been $k+1$ instead of $k+4$, I could have easily see the pattern in which the terms ...
- 758
0
votes
1
answer
38
views
Prove $x_n = \sum_{k=1}^n \frac{1}{(a+(k-1)\cdot d)\cdot(a+k\cdot d)}$ is a bounded sequence.
Let $n \in \mathbb N$ and:
$$
x_n = \sum_{k=1}^n \frac{1}{(a+(k-1)\cdot d)\cdot(a+k\cdot d)}
$$
Prove $\{x_n\}$ is a bounded sequence.
I'm having hard time finishing the proof. Below is what i've ...
- 5,411
2
votes
2
answers
3k
views
Find $x$ if $\frac1{\sin1°\sin2°}+\frac1{\sin2°\sin3°}+\cdots+\frac1{\sin89°\sin90°} = \cot x\cdot\csc x$ [duplicate]
If $$\dfrac1{\sin1°\sin2°}+\dfrac1{\sin2°\sin3°}+\cdots+\dfrac1{\sin89°\sin90°} = \cot x\cdot\csc x$$ and $x\in(0°,90°)$, find $x$.
I tried writing in $\sec$ form but nothing clicked. Any ideas?
1
vote
1
answer
270
views
Compute the values of the Double Sum
Compute the value of the following double sum:
$$\sum_{\mu=1}^n\sum_{\upsilon=\mu+1}^n\frac{\mu^2}{\upsilon(2\upsilon-1)}$$
I started by simply trying to compute the value of the inner sum:
$$\sum_{...
- 2,365
1
vote
4
answers
298
views
Evaluate a sum which almost looks telescoping but not quite:$\sum_{k=2}^n \frac{1}{k(k+2)}$ [duplicate]
Suppose I need to evaluate the following sum:
$$\sum_{k=2}^n \frac{1}{k(k+2)}$$
With partial fraction decomposition, I can get it into the following form:
$$\sum_{k=2}^n \left[\frac{1}{2k}-\frac{1}{...
- 2,365
0
votes
4
answers
186
views
Solving $\left(1+3+5...+(2n+1)\right ) + \left(3.5+5+6.5+...+(\frac{7+3n}{2})\right)=105$ [closed]
$\left(1+3+5...+(2n+1)\right ) + \left(3.5+5+6.5+...+(\frac{7+3n}{2})\right)=105$
It is the equation that I did not understand how to find $n.$
user491034
3
votes
3
answers
153
views
Summing up $3+5+9+17+...$
Find the sum of sum of $3
+5+9+17+...$ till $n$ terms.
Using Method of differences, the sum of the series is
$$\sum\limits_{j=1}^n 2^{j-1}+n$$
I am facing difficulty in evaluating $$\sum\limits_{j=1}...
- 6,111
9
votes
3
answers
369
views
How to evaluate the sum : $\sum_{k=1}^{n} \frac{k}{k^4+1/4}$
I have been trying to figure out how to evaluate the following sum:
$$S_n=\sum_{k=1}^{n} \frac{k}{k^4+1/4}$$
In the problem, the value of $S_{10}$ was given as $\frac{220}{221}$.
I have tried ...
- 671
1
vote
3
answers
325
views
A formula for $1^4+2^4+...+n^4$
I know that
$$\sum^n_{i=1}i^2=\frac{1}{6}n(n+1)(2n+1)$$
and
$$\sum^n_{i=1}i^3=\left(\sum^n_{i=1}i\right)^2.$$
Here is the question: is there a formula for
$$\sum^n_{i=1}i^4.$$
- 4,934
3
votes
2
answers
196
views
Inequality $\frac{1}{a+b}+\frac{1}{a+2b}+...+\frac{1}{a+nb}<\frac{n}{\sqrt{a\left( a+nb \right)}}$
Let $a,b\in \mathbb{R+}$ and $n\in \mathbb{N}$. Prove that:
$$\frac{1}{a+b}+\frac{1}{a+2b}+...+\frac{1}{a+nb}<\frac{n}{\sqrt{a\left( a+nb \right)}}$$
I have a solution using induction, but ...
- 765
6
votes
4
answers
5k
views
$\sum r(r+1)(r+2)(r+3)$ is equal to?
$$\sum r(r+1)(r+2)(r+3)$$ is equal to?
Here, $r$ varies from $1$ to $n$
I am having difficulty in solving questions involving such telescoping series. While I am easily able to do questions where a ...
- 189
1
vote
2
answers
98
views
If $S=\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\cdots+\frac{n}{1+n^2+n^4}$, then calculate $14S$.
If $$S=\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\cdots+\frac{n}{1+n^2+n^4}\,$$ find the value of $14S$.
The question can be simplified to:
Find $S=\sum\limits_{k=1}^n\,t_k$ if $t_n=\dfrac{n}{1+n^2+n^...
- 2,642
0
votes
5
answers
6k
views
Prove $\sum_{k=1}^n \frac{1}{(2k-1)(2k+1)}=\frac{n}{2n+1}$ [closed]
I have attached an image of a kind of mathematical induction question that i have never seen before. I attached it because i don't know how to type all the symbols out properly, i'm sorry again would ...
- 69
6
votes
3
answers
4k
views
How to derive $\sum j^2$ from telescoping property
The book Real Analysis via Sequences and Series has a method of proving that $$\sum_{j=1}^n j = \frac{n(n+1)}{2}$$ that I've never seen before. The way they do it is by starting with $\sum (2j+1)$, ...
- 63
2
votes
2
answers
129
views
Proving that $\sum_{i=1}^n\frac{1}{i^2}<2-\frac1n$ for $n>1$ by induction [duplicate]
Prove by induction that
$1 + \frac {1}{4} + \frac {1}{9} + ... +\frac {1}{n^2} < 2 - \frac{1}{n}$ for all $n>1$
I got up to using the inductive hypothesis to prove that $P(n+1)$ is true but I ...
- 832