Cracking the Series Convergence Puzzle: A Deep Dive into Positive Integers k

Hello, math enthusiasts! Today, we're going to tackle an intriguing problem that'll tickle your brain and test your series convergence skills. So, grab your calculators and let's dive right in! Guys, explore more in Guides And Explainers and for which positive integers k is the following series convergent.

The Mystery Series

We're looking at the following series, which seems to be hiding its convergence behavior from us:

$$1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \dotsb$$

Our mission is to find the positive integers `k` for which this series converges. But first, let's brush up on our series convergence knowledge.

Series Convergence Refresher

Before we dive into the nitty-gritty, let's quickly review two key tests for series convergence:

1. The Divergence Test: If the limit of the `n`-th term as `n` approaches infinity is positive, the series diverges.
2. The Integral Test: If the integral of the function `f(n) = 1/n^k` from 1 to infinity is finite, the series converges.

Now that we're all caught up, let's get back to our series and find those elusive positive integers `k`.

The Pivotal Role of `k`

Our series can be written in the form of a p-series:

$$\sum_{n=1}^{\infty} \frac{1}{n^k}$$

The convergence of p-series depends on the value of `k`. So, let's explore the behavior of our series as we vary `k`.

When `k > 1`

When `k > 1`, the `n`-th term of our series, `1/n^k`, approaches 0 as `n` goes to infinity. This means that the Divergence Test isn't applicable, and we need to turn to the Integral Test.

The integral of `1/n^k` from 1 to infinity is:

$$\in1^\infty \frac{1}{n^k} \,dn = \left[ \frac{1}{k-1}n^{1-k} \right]1^\infty$$

This integral converges if and only if `k - 1 1`. Therefore, our series diverges when `k > 1`.

When `k = 1`

When `k = 1`, our series becomes a harmonic series, which is well-known to diverge. So, `k = 1` is not the answer we're looking for.

When `0

Now, let's consider the case where `0

The Surprising Convergence

After all that, you might be wondering if there's any value of `k` that makes our series converge. The answer lies in the case where `k = 1`.

When `k = 1`, our series becomes a p-series with `p = 1`, which is a special case that converges by the p-series test. So, the only positive integer `k` for which our series converges is `k = 1`.

Wrapping Up

And there you have it, folks! We've successfully unraveled the mystery of the series convergence puzzle. We found that the only positive integer `k` for which the series: