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The task involves applying the theorem for combining series to find the sum of two specific infinite series.

**For series (a):**
We are given the series:
\[
\sum_{k=1}^{\infty} \left(\frac{1}{2k} + \frac{1}{3k}\right)
\]
Using the provided theorem, this can be split into two separate series and their sums:
\[
\sum_{k=1}^{\infty} \left(\frac{1}{2k}\right) + \sum_{k=1}^{\infty} \left(\frac{1}{3k}\right)
\]
This equals:
\[
\frac{1}{2} \sum_{k=1}^{\infty} \left(\frac{1}{k}\right) + \frac{1}{3} \sum_{k=1}^{\infty} \left(\frac{1}{k}\right)
\]
Both sums are the harmonic series multiplied by \(\frac{1}{2}\) and \(\frac{1}{3}\), respectively. However, the harmonic series diverges (\(\sum_{k=1}^{\infty} \frac{1}{k} = \infty\)), therefore:
\[
\frac{1}{2}\infty + \frac{1}{3}\infty = \infty
\]
Thus, series (a) diverges.

**For series (b):**
We have the series:
\[
\sum_{k=1}^{\infty} \left(\frac{1}{5^k} - \frac{1}{k(k+1)}\right)
\]
This is a sum of two different types of series:
- The first term, \(\frac{1}{5^k}\), is a geometric series with \(r = \frac{1}{5}\). Its sum is given by \(\frac{1/5}{1 - 1/5} = \frac{1}{4}\).
- The second term, \(\frac{1}{k(k+1)}\), can be rewritten using partial fractions: \(\frac{1}{k} - \frac{1}{k+1}\). This series telescopes, meaning that most terms cancel out, leaving only the first and the last. However, as \(k\) approaches infinity, the last term approaches zero, leaving the sum as \(1\).

So, series (b) sums to:
\[
\frac{1}{4} - 1 = -\frac{3}{4}
\]

### Answers:
- (a) The series diverges.
- (b) \(-\frac{3}{4}\)

Would you like further details on any part of this explanation or have other questions? Here are some follow-up topics we could discuss:

1. What does it mean for a series to diverge?
2. Can you explain more about geometric series?
3. What is the significance of the harmonic series in mathematics?
4. How does partial fraction decomposition work for series?
5. Can you provide another example of a telescoping series?

**Tip:** When examining a series, always consider its individual terms. Understanding whether each term converges or diverges can greatly simplify your analysis.

Use the theorem for convergent series to find the sum of the series: (a) \( \left( \frac{1}{2} + \frac{1}{3} \right) + \left( \frac{1}{2^2} + \frac{1}{3^2} \right) + ... \) (b) \( \sum_{k=1}^{\infty} \left( \frac{1}{5^k} - \frac{1}{k(k+1)} \right) \).

Learn how to use the theorem of convergent series to solve infinite series problems involving geometric and telescoping series. This guide covers examples like geometric series sums and partial fraction decomposition, suitable for college-level students.

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