All Questions
31
questions
3
votes
4
answers
84
views
Finding and proofing a closed formula for $\sum_{n=1}^k\sqrt{1+\frac{1}{n^2}+\frac{1}{(n+1)^2}}$
I want to find and proof a closed formula for the following sum $$\sum_{n=1}^k\sqrt{1+\frac{1}{n^2}+\frac{1}{(n+1)^2}}=\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\dots +\...
- 524
-1
votes
1
answer
126
views
Find the value $\sum_{n=a}^b\frac1{\sin (2^{n+3})}$ [closed]
Find the value of: $$\sum_{n=0}^{10}\frac1{\sin (2^{n+3})}$$
I'm stuck on this problem, can someone please help?
- 596
-1
votes
1
answer
75
views
how to find the sum of these terms without the gamma function?
While solving a problem based on integration, I arrived at the following
$$\sum\limits_{x = 1}^{38} \ln\left(\frac{x}{x+1}\right)$$
I'm supposed to prove that this is less than $\ln(99)$
in order to ...
- 1,346
2
votes
1
answer
88
views
Best way to solve a summation with binomial coefficients in denominator apart from telecoping method
The value of $\sum_{r=1}^{m}\frac{(m+1)(r-1)m^{r-1}}{r\binom{m}{r}} = \lambda$ then the correct statement is/are
(1) If $m=15$ and $\lambda$ is divided by m then the remainder is 14.
(2) If $m=7$ and $...
- 445
-2
votes
2
answers
243
views
Sum the series : $\frac{1}{9\sqrt11 + 11\sqrt9} +\frac{1}{11\sqrt13 + 13\sqrt11} +\ldots$ [closed]
$$\frac{1}{9\sqrt11 + 11\sqrt9} + \frac{1}{11\sqrt13 + 13\sqrt11} + \frac{1}{13\sqrt15 + 15\sqrt13} + \ldots + \frac{1}{n\sqrt{n+2} + (n+2)\sqrt{n}} = \frac{1}{9}$$
Find the value of $n$.
I got the ...
- 187
3
votes
1
answer
145
views
Find $\sum_{r=1}^{20} (-1)^r\frac{r^2+r+1}{r!}$.
Calculate $$\sum_{r=1}^{20} (-1)^r\frac{r^2+r+1}{r!}\,.$$
I broke the sum into partial fractions and after writing 3-4 terms of the sequence I could see that it cancels but I wasn't able to arrive at ...
- 1,107
2
votes
1
answer
149
views
Evaluate $\sum_{r=1}^{m} \frac{(r-1)m^{r-1}}{r\cdot\binom{m}{r}}$
Evaluate:$$\sum_{r=1}^{m} \frac{(r-1)m^{r-1}}{r\cdot\binom{m}{r}}$$
Using the property:$$r\binom{m}{r}=m\binom{m-1}{r-1}$$
It is same as $$\sum_{r=2}^{m} \frac{(r-1)m^{r-1}}{m\cdot\binom{m-1}{r-1}}$$
...
- 888
0
votes
3
answers
2k
views
If $\sum_{r=0}^{n-1}\log _2\left(\frac{r+2}{r+1}\right)= \prod_{r = 10}^{99}\log _r(r+1)$, then find $n$.
If \begin{align}\sum_{r=0}^{n-1}\log _2\left(\frac{r+2}{r+1}\right) = \prod_{r = 10}^{99}\log _r(r+1).\end{align}
then find $n$.
I found this question in my 12th grade textbook and I just can't wrap ...
- 125
1
vote
1
answer
195
views
Find the sum: $\sum_{n=1}^{20}\frac{(n^2-1/2)}{(n^4+1/4)}$
Hint: this is a telescoping series sum (I have no prior knowledge of partial fraction decomposition)
Attempt: I tried to complete the square but the numerator had an unsimplifiable term. So I couldn't ...
- 45
2
votes
3
answers
625
views
Prove sum of $k^2$ using $k^3$
So the title may be a little bit vague, but I am quite stuck with the following problem.
Asked is to first prove that $(k + 1)^3 - k^3 = 3k^2 + 3k + 1$. This is not the problem however. The question ...
- 181
1
vote
2
answers
83
views
Finding a formula for $\sum_{k=1}^n(k^2-(k-1)^2)$
I have got this following series:
$$\sum_{k=1}^n(k^2-(k-1)^2)$$
I want to come up with a formula for the summation.
I did some math and for me, the formula would be as follows:
$$\sum_{k=1}^n(k^2-(...
- 159
0
votes
2
answers
184
views
Possible telescopic sum
Prove that $$\sum_{k=1}^n 4^{k}\sin^{4} \left(\frac{a}{2^k}\right) = 4^{n}\sin^{2} \left(\frac{a}{2^n}\right) - \sin^{2}a$$
I suspect that telescopic sum is involved but don't know how to proceed. ...
- 407
4
votes
2
answers
392
views
Find the sum of series: $\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{97}+\sqrt{98}}+\frac{1}{\sqrt{99}+\sqrt{100}}$
Find the sum of series:
$$\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\frac{1}{\sqrt{5}+\sqrt{6}}+...+\frac{1}{\sqrt{97}+\sqrt{98}}+\frac{1}{\sqrt{99}+\sqrt{100}}$$
My Attempt:
I tried ...
- 9,599
0
votes
2
answers
52
views
Rewrite $\sum_{n=1}^k\log_3{(\frac{n+1}{n})}$ and write the formula in terms of k
Rewrite $\sum_{n=1}^k\log_3{(\frac{n+1}{n})}$ and write the formula in terms of k.
I rewrote to $1+\frac{1}{n}$ and summed to get (I think) $\log_3(k+\frac{1}{n^k+k!})$ but I'm unsure if the $\log_3$ ...
- 37
0
votes
2
answers
56
views
Rewrite $\sum_{n=1}^k{(n-1)/n!}$ and write the formula in terms of k [closed]
Rewrite $\sum_{n=1}^k{\frac{n-1}{n!}}$
I have turned it into $\frac{1}{n}*\frac{1}{(n-2)!}$ but do not know where to go from here.
- 37
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}$$ that I've never seen before. The way they do it is by starting with $\sum (2j+1)$, ...
- 63
2
votes
2
answers
129
views
Proving that $\sum_{i=1}^n\frac{1}{i^2}<2-\frac1n$ for $n>1$ by induction [duplicate]
Prove by induction that
$1 + \frac {1}{4} + \frac {1}{9} + ... +\frac {1}{n^2} < 2 - \frac{1}{n}$ for all $n>1$
I got up to using the inductive hypothesis to prove that $P(n+1)$ is true but I ...
- 832
15
votes
1
answer
31k
views
Find the sum $\frac{1}{\sqrt{1}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + ...+ \frac{1}{\sqrt{99}+\sqrt{100}}$
I would like to check I have this correct
Find the sum
$$\frac{1}{\sqrt{1}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + ...+ \frac{1}{\sqrt{99}+\sqrt{100}}$$
Hint: rationalise the denominators to ...
- 1,155