Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
39.∑ (k = 2 to ∞) (–0.15)ᵏ
53
views
14. Sequences & Series
Series
3:53 minutesProblem 10.3.7
Textbook Question
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.
Verified step by step guidance1
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.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
06:00
Geometric Series
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.
Recommended video:
08:01Integration Using Partial Fractions
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.
Recommended video:
05:44Algebra Rules for Finite Sums
Related Videos
Related Practice