Find a formula for the nth partial sum Sₙ of
∑ k = 1 to ∞[(1/(k + 3)) − (1/(k + 4))]
Use your formula to find the sum of the first 36 terms of the series.

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Recognize that the series is a telescoping series of the form \( \sum_{k=1}^\infty \left( \frac{1}{k+3} - \frac{1}{k+4} \right) \). This means many terms will cancel out when we write out the partial sums explicitly.

Write the nth partial sum \( S_n \) as \( S_n = \sum_{k=1}^n \left( \frac{1}{k+3} - \frac{1}{k+4} \right) \).

Expand the sum to see the cancellation pattern: \( S_n = \left( \frac{1}{4} - \frac{1}{5} \right) + \left( \frac{1}{5} - \frac{1}{6} \right) + \cdots + \left( \frac{1}{n+3} - \frac{1}{n+4} \right) \). Notice that most intermediate terms cancel out.

After cancellation, the partial sum simplifies to \( S_n = \frac{1}{4} - \frac{1}{n+4} \). This is the formula for the nth partial sum.

To find the sum of the first 36 terms, substitute \( n = 36 \) into the formula: \( S_{36} = \frac{1}{4} - \frac{1}{36+4} \). You can then simplify this expression to get the numerical value.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Telescoping Series

A telescoping series is a series where many terms cancel out when the partial sums are expanded, leaving only a few terms from the beginning and end. This simplification makes it easier to find a formula for the nth partial sum by identifying the pattern of cancellation.

Partial Sums

The nth partial sum, Sₙ, is the sum of the first n terms of a series. Finding a formula for Sₙ helps analyze the behavior of the series and compute sums for specific values of n without adding each term individually.

Evaluating Finite Sums

Once a formula for the nth partial sum is found, it can be used to calculate the sum of a finite number of terms by substituting the desired n value. This process avoids direct summation and leverages the simplified expression derived from the telescoping property.

Find a formula for the nth partial sum Sₙ of∑ k = 1 to ∞[(1/(k + 3)) − (1/(k + 4))]Use your formula to find the sum of the first 36 terms of the series.