query: Proving that $\sum_{i=1}^n\frac{1}{i^2}1$ by induction
positive_passages.docid: 27304
positive_passages.text: Proving $ 1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$ for all $n\geq 2$ by induction
negative_passages.docid: 5272
negative_passages.text: Proving that $\sum_{k=0}^{n}\frac{(-1)^k}{{n\choose k}}=[1+(-1)^n] \frac{n+1}{n+2}.$
negative_passages.docid: 107824
negative_passages.text: Proof by induction that $a_0 = 1, a_1 = 1, a_n=2a_{n-1} + 3a_{n-2}$ satisfies $a_n = \frac12 (3^n) + \frac12 (-1)^n$
negative_passages.docid: 120093
negative_passages.text: Proof of inequality $2(\sqrt{n+1}-\sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n} - \sqrt{n-1})$ using induction
negative_passages.docid: 16776
negative_passages.text: How to prove $\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2}=+\infty$
negative_passages.docid: 16177
negative_passages.text: Proof by Induction: $\sum_0^nx^i=(1-x^{n+1})/(1-x)$
negative_passages.docid: 54595
negative_passages.text: Prove $\sum\limits_{i=2}^{n}\frac{1}{(i-1)i}$ = $\frac{(n-1)}{n}$ using induction.
negative_passages.docid: 33082
negative_passages.text: Proving $\sum_{i=0}^n 2^i=2^{n+1}-1$ by induction.
negative_passages.docid: 79657
negative_passages.text: Prove by induction $\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}$ for $n\ge1$
negative_passages.docid: 44940
negative_passages.text: Proving by induction that $1^3 + 2^3 + 3^3 + \ldots + n^3 = \left[\frac{n(n+1)}{2}\right]^2$
negative_passages.docid: 65315
negative_passages.text: Showing $1+2+\cdots+n=\frac{n(n+1)}{2}$ by induction (stuck on inductive step)
negative_passages.docid: 1125
negative_passages.text: Inductive proof that ${2n\choose n}=\sum{n\choose i}^2.$
negative_passages.docid: 134202
negative_passages.text: Proof by Induction - Math
negative_passages.docid: 11407
negative_passages.text: Proving the summation formula using induction: $\sum_{k=1}^n \frac{1}{k(k+1)} = 1-\frac{1}{n+1}$
negative_passages.docid: 18301
negative_passages.text: Proof by Induction involving Inequality and Factorials as denominators
negative_passages.docid: 110790
negative_passages.text: Prove the following summation inequality: $\sum_{i=1}^n(a_i)\sum_{i=1}^n(\frac{1}{a_i})\ge n^2$
negative_passages.docid: 45914
negative_passages.text: Proving that $n!≤((n+1)/2)^n$ by induction