# Determine whether the following series converges absolutely , conditionally or diverges:

Determine whether the following series converges absolutely , conditionally or diverges:

(i) $\ \sum_{n=0}^{\infty} (-1)^n \frac{k^2-2}{k^2+6} \$

(ii) $\ \sum_{n=0}^{\infty} (-1)^n \frac{k^2}{(3k)!} \$

Answer:

(i)

The series is $\ \sum_{n=0}^{\infty} (-1)^n \frac{k^2-2}{k^2+6} \$

$a_k=(-1)^n \frac{k^2-2}{k^2+6} \$

The series is alternating series .

But the term of series does not decrease by its absolute value. So how can we conclude about the convergence of the series.

further taking absolute value, we get

$\ |a_k|=\frac{k^2-2}{k^2+6} \$

For absolute convergence,

$\lim_{n \to \infty} |\frac{a_{n+1}}{a_n} |=1 \$

so we can say whether converges or not

(ii) The given series is $\ \sum_{n=0}^{\infty} (-1)^n \frac{k^2}{(3k)!} \$

This is an alternatic series test.

The absolute value of each term decreases from the previous term.

Thus by Alternating series test , the series converges.

But the series does not converges absolutely.

Help with the part $\ (i) \$ question

• In (1) we have that $a_n$ does NOT converge to zero. – Robert Z May 8 '18 at 6:48
• Remember that if $\lim\limits_{k\to\infty}a_k\neq 0$ then $\sum\limits_{k=1}^\infty a_k$ does not converge, and vice versa if $\sum\limits_{k=1}^\infty a_k$ does converge then $\lim\limits_{k\to\infty}a_k=0$ – JMoravitz May 8 '18 at 6:53
• So the series diverges in $\ (i) \$ – M. A. SARKAR May 8 '18 at 6:53
• Am I correct about the series in part $\ (ii) \$ ? – M. A. SARKAR May 8 '18 at 6:53
• Why, and how, did you determine the series (ii) does not converge absolutely? – DonAntonio May 8 '18 at 6:58