# Questions tagged [telescopic-series]

For summation questions involving telescopic sums/series. This tag is often used with (summation) or (sequences-and-series).

317 questions
Filter by
Sorted by
Tagged with
155 views

### simplifying $\frac{1}{\sqrt{2} + 1} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + 2}=1$ [duplicate]

$$\frac{1}{\sqrt{2} + 1} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + 2}=1$$ I came across this on a practice standardized test. The question was to evaluate the left hand side, and it ...
• 157
63 views

### AMC 10 Math Question About Formula For A Sequence... [closed]

Find the value of $\frac{{1^2 + 1 \cdot 2 + 2^2}}{{1^3 \cdot 2^3}} + \frac{{2^2 + 2 \cdot 3 + 3^2}}{{2^3 \cdot 3^3}} + \cdots + \frac{{10^2 + 10 \cdot 11 + 11^2}}{{10^3 \cdot 11^3}}$ This is a ...
57 views

• 108k
73 views

• 524
76 views

46 views

### Help for Telescopic Riemann sum

Consider the Riemann sum $$\sum_{k=1}^n 2x^∗_k ∆x_k$$ of the integral of f(x) = 2x in an interval [a, b]. (a) Show that if $$x^∗_k$$ is the midpoint of the k−th subinterval, then the Riemann sum is ...
1k views

### Is every series a telescoping series?

This question may seem silly at first. We say that a series $\sum a_n$ is a telescoping series if there exists a sequence $(b_n)$ with $a_n=b_n-b_{n+1}$ for every $n$. One can show that $\sum a_n$ ...
• 1,146
43 views

### 1 + 1/2 * 1/3 + (1 * 3)/(2 * 4) * 1/5 + (1 * 3 * 5)/(2 * 4 * 6) * 1/7 + .. converges [duplicate]

To prove that $1+ \frac{1}{2} \frac{1}{3}+\frac{1}{2} \frac{3}{4} \frac{1}{5} + \frac{1}{2} \frac{3}{4} \frac{5}{6}\frac{1}{7}+....$ converges. If i attempt to find the nth term for this series i ...
108 views

### Does $\sum_{r = 1}^n \ln\left(\frac{1 + r}{r}\right) = \ln (\Gamma(n + 2)) - \ln (\Gamma(n+1))$? If so, why?

When attempting the evaluate the integral $\int_0^1 \{\ln(x)\}$, where $\{ x \}$ is the fractional part function, I came across the following sum: $$\sum_{r = 1}^n \ln\left(\frac{r + 1}{r}\right)$$ ...
• 47
1 vote
70 views

• 16.4k
65 views

61 views

• 305
73 views

• 387
113 views

### Using Gosper's algorithm to obtain the WZ certificate of $\sum \binom{n}{k} = 2^n$

I'm not sure where my work is wrong, I'm not obtaining an answer, even though I know there should be one. In order to obtain the WZ proof certificate for the sum $$\sum_{k=0}^n \binom{n}{k} = 2^n$$ ...
• 2,571
### What is the value of $a_1a_2\cdots a_{2019}$?
Let $a_1=\frac 34$ and for any $n\geq2$ $4a_n=4a_{n-1}+\frac {2n+1}{1^3+2^3+\cdots n^3 }$. What is the value of $a_1a_2\cdots a_{2019}$? I tried $1^3+2^3+\cdots +n^3=\frac {n^2(n+1)^2}{4}$ and I ...