The document you uploaded contains a calculus problem set related to series convergence. Here’s a breakdown of the tasks presented:

1. Problem 2 asks to prove the convergence of various series and to show that their sums are as indicated. This involves evaluating whether each series converges and calculating the sum if it does.

2. Problem 3 requires determining the convergence or divergence of several series and justifying the answer. For each series, you’ll likely need to apply convergence tests, such as the Ratio Test, Root Test, Comparison Test, or Alternating Series Test, depending on the form of each series.

Would you like me to go through a specific part of the problem set, such as one of the series from Problem 2 or Problem 3? Or, if you prefer, I can provide a summary of convergence tests that might help you solve these problems.

Here are five follow-up questions based on the content:

1. Would you like detailed help with the convergence tests required for each series in Problem 2?
2. Do you need guidance on identifying the most suitable convergence test for each series in Problem 3?
3. Are there any specific series you would like to analyze first?
4. Would you like explanations of common series convergence concepts, such as the Ratio or Root Test?
5. Would you prefer a step-by-step solution for one of the series as a detailed example?

Tip: When analyzing series for convergence, start by checking if they resemble a familiar type, like a geometric or p-series, as this can simplify your choice of test.