Prove by induction $\sum _{i=1}^n\left(-1\right)^{i+1}\:\binom{n}{i}\:\frac{1}{i}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}$
5332
Find the value of $\sum_0^n \binom{n}{k} (-1)^k \frac{1}{k+1}$
16177
Proof by Induction: $\sum_0^nx^i=(1-x^{n+1})/(1-x)$
145274
Prove by induction $\sum \frac {1}{2^n} < 1$
33082
Proving $\sum_{i=0}^n 2^i=2^{n+1}-1$ by induction.
95085
Prove by induction $ \sin x + \sin 2x + ... + \sin nx = \frac {\sin (\frac {n + 1} {2} x)} {\sin \frac{x}{2}} \sin \frac{nx}{2} $
63474
Showing $\sum_{i = 1}^n\frac1{i(i+1)} = 1-\frac1{n+1}$ without induction?
114212
How to use Mathematical Induction to prove $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \cdots + \frac{1}{n(n + 1)} = \frac{n}{n + 1}$?
72639
Show that $\sum_{k=0}^n\binom{3n}{3k}=\frac{8^n+2(-1)^n}{3}$
49066
Give the combinatorial proof of the identity $\sum\limits_{i=0}^{n} \binom{k-1+i}{k-1} = \binom{n+k}{k}$
20271
Proving $\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$ without induction
131565
Proof by induction of summation inequality: $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\frac1{2^n}\ge 1+\frac{n}2$
108596
Proof by induction for "sum-of"
130543
If $a_n = \sum\limits_{r=0}^n \binom{n}{r} b_r$, prove $(-1)^n b_n = \sum\limits_{s=0}^n \binom{n}{s} (-1)^s a_s$
87941
Prove by mathematical induction that $\forall n \in \mathbb{N} : \sum_{k=1}^{2n} \frac{(-1)^{k+1}}{k} = \sum_{k=n+1}^{2n} \frac{1}{k} $
133905
Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$
58247
Prove $\sum_{k=0}^n \binom{n+k}{k}=\binom{2n+1}{n}$
112907
Math Induction Proof: $(1+\frac1n)^n < n$