Math / Calculus

Questionwhat number it converges to. a) $\sum_{n=1}^{\infty} \cos \left(\frac{1}{n}\right)-\cos \left(\frac{1}{n+2}\right)$

Studdy Solution

STEP 1

1. The series given is telescoping, which means many terms will cancel each other out.
2. We need to identify the pattern of cancellation to find the limit of the series.
3. We assume the series converges, and we will find the sum it converges to.

STEP 2

1. Identify the general term of the series.
2. Analyze the telescoping nature of the series.
3. Determine the first few terms and the pattern of cancellation.
4. Calculate the limit of the remaining terms as $n \to \infty$ .

STEP 3

Identify the general term of the series:
The series is given by:
$\sum_{n=1}^{\infty} \left( \cos \left(\frac{1}{n}\right) - \cos \left(\frac{1}{n+2}\right) \right)$
This is a difference of cosines.

STEP 4

Analyze the telescoping nature of the series:
Notice that each term has the form:
$\cos \left(\frac{1}{n}\right) - \cos \left(\frac{1}{n+2}\right)$
This suggests that terms will cancel when expanded.

STEP 5

Determine the first few terms and the pattern of cancellation:
Write out the first few terms:
- For $n = 1$ : $\cos(1) - \cos\left(\frac{1}{3}\right)$ - For $n = 2$ : $\cos\left(\frac{1}{2}\right) - \cos\left(\frac{1}{4}\right)$ - For $n = 3$ : $\cos\left(\frac{1}{3}\right) - \cos\left(\frac{1}{5}\right)$
Notice that terms like $\cos\left(\frac{1}{3}\right)$ cancel out.

STEP 6

Calculate the limit of the remaining terms as $n \to \infty$ :
After cancellation, the remaining terms are:
$\cos(1) + \cos\left(\frac{1}{2}\right) - \lim_{n \to \infty} \cos\left(\frac{1}{n+1}\right) - \cos\left(\frac{1}{n+2}\right)$
As $n \to \infty$ , both $\cos\left(\frac{1}{n+1}\right)$ and $\cos\left(\frac{1}{n+2}\right)$ approach $\cos(0) = 1$ .
Thus, the series converges to:
$\cos(1) + \cos\left(\frac{1}{2}\right) - 1 - 1 = \cos(1) + \cos\left(\frac{1}{2}\right) - 2$
The series converges to $\boxed{\cos(1) + \cos\left(\frac{1}{2}\right) - 2}$ .

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Questionwhat number it converges to. a) $\sum_{n=1}^{\infty} \cos \left(\frac{1}{n}\right)-\cos \left(\frac{1}{n+2}\right)$

Studdy Solution

STEP 1

1. The series given is telescoping, which means many terms will cancel each other out.
2. We need to identify the pattern of cancellation to find the limit of the series.
3. We assume the series converges, and we will find the sum it converges to.

STEP 2

1. Identify the general term of the series.
2. Analyze the telescoping nature of the series.
3. Determine the first few terms and the pattern of cancellation.
4. Calculate the limit of the remaining terms as $n \to \infty$ .

STEP 3

Identify the general term of the series:
The series is given by:
$\sum_{n=1}^{\infty} \left( \cos \left(\frac{1}{n}\right) - \cos \left(\frac{1}{n+2}\right) \right)$
This is a difference of cosines.

STEP 4

Analyze the telescoping nature of the series:
Notice that each term has the form:
$\cos \left(\frac{1}{n}\right) - \cos \left(\frac{1}{n+2}\right)$
This suggests that terms will cancel when expanded.

STEP 5

STEP 6

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Questionwhat number it converges to. a) ∑n=1∞cos⁡(1n)−cos⁡(1n+2)\sum_{n=1}^{\infty} \cos \left(\frac{1}{n}\right)-\cos \left(\frac{1}{n+2}\right)∑n=1∞​cos(n1​)−cos(n+21​)

Studdy Solution

STEP 1

1. The series given is telescoping, which means many terms will cancel each other out.2. We need to identify the pattern of cancellation to find the limit of the series.3. We assume the series converges, and we will find the sum it converges to.

STEP 2

1. Identify the general term of the series.2. Analyze the telescoping nature of the series.3. Determine the first few terms and the pattern of cancellation.4. Calculate the limit of the remaining terms as n→∞ n \to \infty n→∞.

STEP 3

Identify the general term of the series:The series is given by:∑n=1∞(cos⁡(1n)−cos⁡(1n+2)) \sum_{n=1}^{\infty} \left( \cos \left(\frac{1}{n}\right) - \cos \left(\frac{1}{n+2}\right) \right) n=1∑∞​(cos(n1​)−cos(n+21​))This is a difference of cosines.

STEP 4

Analyze the telescoping nature of the series:Notice that each term has the form:cos⁡(1n)−cos⁡(1n+2) \cos \left(\frac{1}{n}\right) - \cos \left(\frac{1}{n+2}\right) cos(n1​)−cos(n+21​)This suggests that terms will cancel when expanded.

STEP 5

STEP 6

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Questionwhat number it converges to. a) $\sum_{n=1}^{\infty} \cos \left(\frac{1}{n}\right)-\cos \left(\frac{1}{n+2}\right)$

1. The series given is telescoping, which means many terms will cancel each other out.
2. We need to identify the pattern of cancellation to find the limit of the series.
3. We assume the series converges, and we will find the sum it converges to.

1. Identify the general term of the series.
2. Analyze the telescoping nature of the series.
3. Determine the first few terms and the pattern of cancellation.
4. Calculate the limit of the remaining terms as $n \to \infty$ .

Identify the general term of the series:
The series is given by:
$\sum_{n=1}^{\infty} \left( \cos \left(\frac{1}{n}\right) - \cos \left(\frac{1}{n+2}\right) \right)$
This is a difference of cosines.

Analyze the telescoping nature of the series:
Notice that each term has the form:
$\cos \left(\frac{1}{n}\right) - \cos \left(\frac{1}{n+2}\right)$
This suggests that terms will cancel when expanded.