What does the sum $0(1/4)+1(3/4)(1/4)+2(3/4)^2(1/4)+3(3/4)^3(1/4)+⋯$ equal?
6416
How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$?
55061
Evaluating the sum after reordering an infinite series $1-\frac {1} {2}+\frac {1} {3}-\frac {1} {4}+\ldots $
104542
Evaluate $\frac{1}{3}+\frac{1}{4}\frac{1}{2!}+\frac{1}{5}\frac{1}{3!}+\dots$
79657
Prove by induction $\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}$ for $n\ge1$
79170
$1 + \frac{1}{1+2}+ \frac{1}{1+2+3}+ ... + \frac{1}{1+2+3+...+n} = ?$
50606
Find $\sum\limits_{k\, \text{ odd}} \frac{2(k^2-1)}{k^4+k^2+1}$
40784
$\sum_{k=m}^{n}(-1)^k\binom{n}{k}\binom{k}{m}=0, n>m\geq 0$
14399
Calculate $\sum_{k=1}^n (-1)^{k+1} \binom{n}{k}\frac{1}{k}$
111602
Help with telescoping sum $\sum_{i=3}^n \frac{1}{i(i+3)} $
123820
Guess the formula for $\sum\frac 1{(4n-3)(4n+1)}$ and prove by induction
73570
Sum of (example: 0+1+2+3 = 6 , 0+1+2+3+4+5+6+7 = 28 and so on)
89698
Please help with $\sum\limits_{r=1}^{N}\frac{2r+3}{3^r(r+1)}$
78473
Showing $ \sum_{n=0}^{\infty} \frac{1}{(3n+1)(3n+2)}=\frac{\pi}{3\sqrt{3}}$
133076
Prove this sum equal to an expression.
58262
Find the sum of the series $1+\frac{1}{3}\cdot\frac{1}{4}+\frac{1}{5}\cdot\frac{1}{4^2}+\frac{1}{7}\cdot\frac{1}{4^3}+\cdots$
32637
What is the sum of this series: $\displaystyle\sum_{n=0}^{\infty} \dfrac{a^n}{(3n)!}$
62683
How does the sum of the series “$1 + 2 + 3 + 4 + 5 + 6\ldots$” to infinity = “$-1/12$”?