Examples from Infinite series All examples from Infinite series.
[ EXAMPLE ] For r = 1 / 2 r=1/2 r = 1/2
∑ n = 1 ∞ 1 2 n = 1 2 + 1 4 + 1 8 + ⋯ = 1 / 2 1 − 1 / 2 = 1. \sum_{n=1}^{\infty}\frac1{2^n}
=\frac12+\frac14+\frac18+\cdots
=\frac{1/2}{1-1/2}=1. n = 1 ∑ ∞ 2 n 1 = 2 1 + 4 1 + 8 1 + ⋯ = 1 − 1/2 1/2 = 1. [RIGHT CLICK] [ASK AI] [ EXAMPLE ] Since
k ( k + 1 ) ! = 1 k ! − 1 ( k + 1 ) ! , \frac{k}{(k+1)!}
=\frac1{k!}-\frac1{(k+1)!}, ( k + 1 )! k = k ! 1 − ( k + 1 )! 1 , we have
∑ k = 1 N k ( k + 1 ) ! = 1 − 1 ( N + 1 ) ! ⟶ 1. \sum_{k=1}^{N}\frac{k}{(k+1)!}
=1-\frac1{(N+1)!}
\longrightarrow1. k = 1 ∑ N ( k + 1 )! k = 1 − ( N + 1 )! 1 ⟶ 1. Therefore
∑ k = 1 ∞ k ( k + 1 ) ! = 1. \sum_{k=1}^{\infty}\frac{k}{(k+1)!}=1. k = 1 ∑ ∞ ( k + 1 )! k = 1. [RIGHT CLICK] [ASK AI] [ EXAMPLE ] Because
2 n 3 n + 1 ⟶ 2 3 ≠ 0 , \frac{2n}{3n+1}\longrightarrow\frac23\ne0, 3 n + 1 2 n ⟶ 3 2 = 0 , the series
∑ n = 0 ∞ 2 n 3 n + 1 \sum_{n=0}^{\infty}\frac{2n}{3n+1} n = 0 ∑ ∞ 3 n + 1 2 n diverges.
[RIGHT CLICK] [ASK AI] [ EXAMPLE ] Although
1 n ⟶ 0 , \frac1n\longrightarrow0, n 1 ⟶ 0 , the harmonic series diverges. Group its terms:
1 + 1 2 + ( 1 3 + 1 4 ) + ( 1 5 + ⋯ + 1 8 ) + ⋯ . 1+\frac12
+\left(\frac13+\frac14\right)
+\left(\frac15+\cdots+\frac18\right)+\cdots. 1 + 2 1 + ( 3 1 + 4 1 ) + ( 5 1 + ⋯ + 8 1 ) + ⋯ . Every block after the first has total at least 1 / 2 1/2 1/2 2 j − 1 + 1 2^{j-1}+1 2 j − 1 + 1 2 j 2^j 2 j 2 j − 1 2^{j-1} 2 j − 1 1 / 2 j 1/2^j 1/ 2 j 1 / 2 1/2 1/2
[RIGHT CLICK] [ASK AI] Infinite series Builds on Sequences and limits .