Math Snap

STEP 1

1. The series given is a telescoping series.
2. A telescoping series is one where most terms cancel out, leaving a finite number of terms.
3. We need to determine if the series converges and, if so, find the sum.

STEP 2

1. Identify the nature of the series.
2. Analyze the series to determine convergence.
3. Calculate the sum of the series if it converges.

STEP 3

Identify the nature of the series. The series is given by:
$$\sum_{k=6}^{\infty}\left[\frac{1}{k}-\frac{1}{k+1}\right]$$ This is a telescoping series because each term can be written as the difference of two fractions, and many terms will cancel each other out.

STEP 4

To determine convergence, examine the partial sums of the series. The $n$-th partial sum $S_n$ is:
$$S_n = \sum_{k=6}^{n}\left[\frac{1}{k}-\frac{1}{k+1}\right]$$ Write out the first few terms to see the pattern:
$$S_n = \left(\frac{1}{6} - \frac{1}{7}\right) + \left(\frac{1}{7} - \frac{1}{8}\right) + \ldots + \left(\frac{1}{n} - \frac{1}{n+1}\right)$$ Notice that all intermediate terms cancel out, leaving:
$$S_n = \frac{1}{6} - \frac{1}{n+1}$$ As $n \to \infty$, the term $\frac{1}{n+1} \to 0$.

STEP 5

Since the limit of the partial sum $S_n$ as $n \to \infty$ exists and is finite, the series converges.
$$\lim_{n \to \infty} S_n = \frac{1}{6} - 0 = \frac{1}{6}$$

Since the series converges, the sum of the series is:
$$\sum_{k=6}^{\infty}\left[\frac{1}{k}-\frac{1}{k+1}\right] = \frac{1}{6}$$ The series converges, and the sum is:
$$\boxed{\frac{1}{6}}$$