Math / Calculus

QuestionGiven the series: $\sum_{k=6}^{\infty}\left[\frac{1}{k}-\frac{1}{k+1}\right]$ does this series converge or diverge? converges diverges 0
If the series converges, find the sum of the series: $\sum_{k=6}^{\infty}\left[\frac{1}{k}-\frac{1}{k+1}\right]=$ $\square$ (If the series diverges, just leave this second box blank.) Question Help: Video Message instructor

Studdy Solution

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.
$\lim_{n \to \infty} S_n = \frac{1}{6} - 0 = \frac{1}{6}$

STEP 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:
$\boxed{\frac{1}{6}}$

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Studdy Solution

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

STEP 5

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

STEP 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:
$\boxed{\frac{1}{6}}$

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Studdy Solution

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

STEP 4

STEP 5

Since the limit of the partial sum Sn S_n Sn​ as n→∞ n \to \infty n→∞ exists and is finite, the series converges.lim⁡n→∞Sn=16−0=16 \lim_{n \to \infty} S_n = \frac{1}{6} - 0 = \frac{1}{6} n→∞lim​Sn​=61​−0=61​

STEP 6

Since the series converges, the sum of the series is:∑k=6∞[1k−1k+1]=16 \sum_{k=6}^{\infty}\left[\frac{1}{k}-\frac{1}{k+1}\right] = \frac{1}{6} k=6∑∞​[k1​−k+11​]=61​ The series converges, and the sum is:16 \boxed{\frac{1}{6}} 61​​

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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.

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

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:
$\boxed{\frac{1}{6}}$