Prove that the given series diverges by showing that the N^th partial sum satisfies SN ≥k ·N fo

step 1 Hello, so here we have that s sub n is going to be the partial sum, where so s of n is equal to

Prove that the given series diverges by showing that the...

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Key Concepts

Divergence of Infinite Series

An infinite series diverges if its sequence of partial sums does not converge to a finite limit. One common method to show divergence is to prove that, beyond a certain point, the partial sums exceed any given bound. Demonstrating that the Nth partial sum grows at least linearly (e.g., S? ? k·n for some k > 0) confirms that the series cannot settle to a finite value, leading to divergence.

Partial Sum Analysis

Partial sum analysis involves summing the first N terms of a series to understand its overall behavior. In this context, obtaining a lower bound on the partial sums is key; if the lower bound grows without limit (for example, linearly with N), then the entire series diverges. This technique helps in making explicit the unbounded growth of the series rather than relying solely on the convergence of individual terms.

Comparison Test

The comparison test is a method for determining the convergence or divergence of a series by comparing it with another series whose behavior is known. In the problem at hand, the given series is compared to a simpler series (or bound) that grows linearly. Establishing that the given series’ terms are bounded below by a constant multiple of the terms of a divergent series directly implies the divergence of the original series.

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