Show that sum $1+z+z^2+...+z^{n-1}=0$.
90043
Show that if $r$ is an nth root of $1$ and $r\ne1$, then $1 + r + r^2 + ... + r^{n-1} = 0$.
53866
Prove $ x^n-1=(x-1)(x^{n-1}+x^{n-2}+...+x+1)$
122534
Show that $\sum (-1)^{n+1}/n$ = $k$ can be expanded to a sum with $ 1 - 1/2 + 1/3 + 1/5 - 1/4 +1/7 + 1/9 - 1/6 + ....$ which is $3k/2$
1167
How to prove that $\sum_{k=0}^{n}(-1)^k \binom {n}{k}=0 $
6539
Prove $\sum_{i=1}^n i! \cdot i = (n+1)! - 1$?
37650
Find $\sum_{m=0}^n\ (-1)^m m^n {n \choose m}$
108244
Proof without solving equation that if $(1+z)^n = (1-z)^n$ then Re(z) = 0. Did I do it right?
91458
Sum of the series $\binom{n}{0}-\binom{n-1}{1}+\binom{n-2}{2}-\binom{n-3}{3}+..........$
71162
Showing that $\sum_{n=1}^{\infty} \frac{z^n}{1+z^{2n}}$ converges
106552
Find the sum of $\sum \frac{1}{k^2 - a^2}$ when $0<a<1$
63474
Showing $\sum_{i = 1}^n\frac1{i(i+1)} = 1-\frac1{n+1}$ without induction?
66568
Number Theory : Show that $\sum_{i=0}^\infty$ $ [\frac{n}{2^i}+\frac{1}{2}]$ $=$ $2n$
148442
Show that the sum $\sum_{k = 0}^n 2^k \binom{n}{k}$ is equal to $3^n$
15943
show $\sum_{cyc}(1-x)^2\ge \sum_{cyc}\frac{z^2(1-x^2)(1-y^2)}{(xy+z)^2}.$
113189
Find the sum of the series $1^2-2^2+3^2-4^2+...-(2n)^2$
120976
Proving $\sum_{k=0}^{n}k{n\choose k}^2 = n{2n-1 \choose n-1} $
58247
Prove $\sum_{k=0}^n \binom{n+k}{k}=\binom{2n+1}{n}$