query: Prove that $\sum_{k=1}^{n}(-1)^{k+1}\frac{1}{k}\binom{n}{k} = 1+\frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}$
positive_passages.docid: 20317
positive_passages.text: Proving $\sum_{k=1}^{n}{(-1)^{k+1} {{n}\choose{k}}\frac{1}{k}=H_n}$
negative_passages.docid: 14064
negative_passages.text: Proof by induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$
negative_passages.docid: 4327
negative_passages.text: proving $\left(1+\frac 1n\right)^{n} = 1 + \sum_{k=1}^n \{\frac 1{k!}\prod_{r=0}^{k-1}(1-\frac rn)\}$ using the binomial theorem
negative_passages.docid: 30672
negative_passages.text: Show that $\sum\limits_{k=0}^n\binom{2n}{2k}^{\!2}-\sum\limits_{k=0}^{n-1}\binom{2n}{2k+1}^{\!2}=(-1)^n\binom{2n}{n}$
negative_passages.docid: 19571
negative_passages.text: Combinatorial proof for $\sum_{k = 0}^n \binom {r+k} k = \binom {r + n + 1} n$
negative_passages.docid: 130543
negative_passages.text: If $a_n = \sum\limits_{r=0}^n \binom{n}{r} b_r$, prove $(-1)^n b_n = \sum\limits_{s=0}^n \binom{n}{s} (-1)^s a_s$
negative_passages.docid: 92440
negative_passages.text: Let $n \in \mathbb{Z}^+$, prove the identity $ \sum_{k=1}^{n-1} \binom {n} {k} \frac{kn^{n-k}}{k+1}=\frac{n(n^{n}-1)}{n+1}$
negative_passages.docid: 122534
negative_passages.text: 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 $3k/2$
negative_passages.docid: 89728
negative_passages.text: Prove that $1<\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1}$
negative_passages.docid: 131400
negative_passages.text: Solution check: Let $f_n$ be fibonacci numbers. Prove: $\sum_{k=0}^{n-1} \binom{n+k}{2k+1} = f_{2n-1}$ and $\sum_{k=0}^n \binom{n+k}{2k} = f_{2n}$
negative_passages.docid: 1167
negative_passages.text: How to prove that $\sum_{k=0}^{n}(-1)^k \binom {n}{k}=0 $
negative_passages.docid: 72639
negative_passages.text: Show that $\sum_{k=0}^n\binom{3n}{3k}=\frac{8^n+2(-1)^n}{3}$
negative_passages.docid: 128309
negative_passages.text: Use Mathematical Induction to prove that $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} +...+\frac{1}{n(n+1)}=1-\frac{1}{n+1}$
negative_passages.docid: 76319
negative_passages.text: How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$?
negative_passages.docid: 56138
negative_passages.text: Proving $\sum_{k=0}^{2m}(-1)^k{\binom{2m}{k}}^3=(-1)^m\binom{2m}{m}\binom{3m}{m}$ (Dixon's identity)
negative_passages.docid: 108401
negative_passages.text: Sum of binomial coefficients $\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{2n - 2k}{n - 1} = 0$
negative_passages.docid: 64378
negative_passages.text: Proving a binomial sum identity $\sum _{k=0}^n \binom nk \frac{(-1)^k}{2k+1} = \frac{(2n)!!}{(2n+1)!!}$