Answer to Question #266793 in Calculus for Sayem

Question

Determine for which value of $\alpha$ the given series converge absolutely, converge conditionally, or diverges.

1. $\displaystyle\sum_{k=1}^ ∞ ((-1)$ ^k+1 * $\alpha^k$ ) / k²
2. $\displaystyle\sum_{k=1}^ ∞ ((\alpha -1)^k ​$ ) / k

For each of the following series, determine whether it is absolutely convergent, conditionally convergent, or divergent. Justify your answer.

1. $\displaystyle\sum_{n=2}^ ∞ (-1)^n/(n ln n)$

Answer 1

1.

series diverges if

$\displaystyle{\lim_{k\to \infin}}|a_k|=\displaystyle{\lim_{k\to \infin}}|\alpha^k/k^2|=\infin\implies |\alpha|>1$

series converge absolutely if

$\displaystyle{\lim_{k\to \infin}}|a_k|=\displaystyle{\lim_{k\to \infin}}|\alpha^k/k^2|=0\implies |\alpha|<1$

2.

series diverges if

$\displaystyle{\lim_{k\to \infin}}|a_k|=\displaystyle{\lim_{k\to \infin}}|(\alpha-1)^k/k|=\infin\implies |\alpha|>1$

series converge absolutely if

$\displaystyle{\lim_{k\to \infin}}|a_k|=\displaystyle{\lim_{k\to \infin}}|(\alpha-1)^k/k|=0\implies |\alpha-1|<1\implies 0<\alpha<1$ and $1<\alpha<2$

3.

for alternating series:

series converges if

$|a_n|$ decreases monotonically, i. e. $|a_{n+1}|\le |a_n|$ and $\displaystyle{\lim_{n\to \infin}}|a_n|=0$

$\displaystyle{\lim_{n\to \infin}}|a_n|=\displaystyle{\lim_{n\to \infin}}|1/(nlnn)|=0$

$\frac{1}{nlnn}\ge\frac{1}{(n+1)ln(n+1)}$

so, series converge

for series of absolute values $\displaystyle\sum_{n=2}^{\infin}\frac{1}{nlnn}$ :

$\int_2^{\infin}\frac{1}{xlnx}dx=ln(lnx)|^{\infin}_2=\infin$

so, $\displaystyle\sum_{n=2}^{\infin}\frac{1}{nlnn}$ diverges, this means that $\displaystyle\sum_{n=2}^{\infin}\frac{(-1)^n}{nlnn}$ converge conditionally