Consider two series \(\mathop \sum \limits_{n = 1}^\infty {a_n}\)

SOLUTION

Concept:

A p-series is a specific type of infinite series. It's a series of the form as shown below,

\(\mathop \sum \nolimits_{n = 1}^\infty \frac{1}{{{n^p}}} = \frac{1}{{{1^p}}}+\frac{1}{{{2^p}}}+\frac{1}{{{3^p}}}+... \)

With p-series,

If p > 1, the series will converge, If 0 < p ≤ 1, the series will diverge.

The Ratio test:

Theorem: Let ∑an be a series and If \(\mathop {\lim }\limits_{n \to \infty } \frac{{{a_{k + 1}}}}{{{a_k}}} = L\), and

if L < 1 then the series converges, but if L > 1 the series diverges.

If L = 1, then the test is inconclusive, then we have to go for another test.

Explanation:

Consider the first series:

 \(\mathop \sum \limits_{n = 1}^\infty {a_n} \Rightarrow \mathop \sum \limits_{n = 1}^\infty \frac{1}{{n\sqrt n }} \Rightarrow \mathop \sum \limits_{n = 1}^\infty \frac{1}{{{n^{\frac{3}{2}}}}}\)

This takes the form of p series \(\mathop \sum \limits_{n = 1}^\infty \frac{1}{{{n^p}}}\)

Here p = 3/2

p > 1 ⇒ Convergent

Consider the second series \(\mathop \sum \limits_{n = 1}^\infty {b_n} \Rightarrow \mathop \sum \limits_{n = 1}^\infty \frac{1}{{n!}}\)

By ratio test, \( \Rightarrow \mathop {\lim }\limits_{n \to \infty } \frac{{{b_{n + 1}}}}{{{b_n}}} = {\rm{lim}}\frac{1}{{n + 1}} = 0\)

Here L = 0 (< 1) ⇒ Convergent

Therefore, both the series are convergent.