If $z^2 = \sum_{k=1}^{n}{z_k^2}$, then $|\text{Re}z| \le \sum_{k=1}^{n}{|\text{Re} z_k|}$
90549
If $z^2 = z_1^2+z_2^2+z_3^2$ then $|\Re(z)| \le|\Re(z_1)|+|\Re(z_2)|+|\Re(z_3)|$
94962
For complex $z$, $|z| = 1 \implies \text{Re}\left(\frac{1-z}{1+z}\right)  = 0$
155005
How to show $\sum_{k=n}^\infty{\frac{1}{k!}} \leq \frac{2}{n!}$
72313
Show that for all real numbers $a$ and $b$, $\,\, ab \le (1/2)(a^2+b^2)$
1525
If for any $k$, $\sum\limits_{n=0}^\infty a_n^k=\sum\limits_{n=0}^\infty b_n^k$, then$(a_n)=(b_{σ(n)}),\ σ \in{\mathfrak S}_{\mathbb N}$
127047
How to show that $\sum_{i=1}^n | \langle f, f_i\rangle |^2 \leq \Vert f \Vert^2$
30518
Formula for $\sum_{k=1}^n \frac{1}{k(k+1)(k+2)}$?
50972
Prove that $\sum_{x=1}^{n} \frac{1}{x (x+1)(x+2)} = \frac{1}{4} - \frac{1}{2 (n+1) (n+2)}$
35184
If $z_n \to z$ then $(1+z_n/n)^n \to e^z$
36083
If $k > 1$, then $\frac1{(k - 1)^2}-\frac1{(k + 1)^2}=\frac{4k}{(k^2 - 1)^2}$, hence simplify $\sum\limits_{k = 2}^n\frac k{(k^2 - 1)^2}$
13291
Proving $\sum_{k=1}^n\frac1{\sqrt k}<2\sqrt n$ by induction
69921
Prove that: $(1+a_1)(1+a_2)...(1+a_n) \le 1 + S_n + (S_n)^2/2! + ... + (S_n)^n/n!$
48251
Prove that if $n \in \mathbb{N}$, then $\sum_{d|n}{(d(n))^3}=(\sum_{d|n}{d(n)})^2$ where $d(n)$ is the divisor function.
10917
Give a combinatorial proof: $ n(n+1)2^{n-2} = \sum_{k=1}^{n}k^2\binom{n}{k} $
134812
Proving that if $\sum\|f_n-e_n\|^2< 1$, $\{f_n\}$ is a complete sequence
139074
Question about $a_n=\sum_{k=1}^n \frac{n}{n^2+k}\ $for $ n \in\mathbb{N}$
103840
Calculate $\sum_{k=1}^n k2^{n-k}$