The series \(\sum {\left( {\frac{1}{{np}}} \right)} \) is divergent if

Concept:

Convergence and divergence of an Infinite series.

This is dependent on the convergence (or) divergence of the sequence of partial sums.

Let \(\displaystyle\sum_{k=1}^\infty U_k bc\) an in fenite series.

{S_n} be the sequence of partial sums.

Case (1)

If \(\displaystyle\lim_{n \rightarrow \infty} S_n = S\), Where 'S' is a real number, then the infinite series is converges and \(\displaystyle\sum_{k=1}^\infty U_K =S\)

Case (2)

If \(\displaystyle\lim_{n\rightarrow \infty} S_n\) doesn't have a fenite limit, then series diverges.

If we can determine the whether limit of partial sum of an infinite series is fenite, then we know the infinite series converges else not.

Based on above definitions below table gives the common and recognizable infinite series.

Conclusion:

The Given series is p-series, So, it's divergent if p ≤ 1.

let p = -2

\(\displaystyle\sum_{n=1}^\infty \dfrac{1}{n^{-2}} = \sum_{n=1}^\infty n^2 = 0 + 1 + 4 + ... \infty\)   not finite value.