A friendly warm-up. Consider the geometric series ∑k=0^∞ r^k, where the real number r gives the

step 1 This question will consider the geometric series of the form so much of the r power the r power

A friendly warm-up. Consider the geometric series ∑k=0^∞...

A friendly warm-up. Consider the geometric series $\sum_{k=0}^{\infty} r^k$, where the real number $r$ gives the ratio between successive terms. Note that the series starts with $k=0$ rather than $k=1$. a. Let $r=\frac{4}{5}$ and use the computer to investigate the partial sums of the series $\sum_{k=0}^{\infty}\left(\frac{4}{5}\right)^k$. As you evaluate partial sums with more and more terms, what appears to be the limiting value? Confirm that this value for the sum of the infinite series agrees with the well-known formula for the sum of a geometric series whose ratio $r$ has absolute value less than 1 . b. Let $r=1.01$ and use the computer to study the partial sums. What do you conclude about the convergence or divergence of the geometric series in this case? Explain. c. Now let $r=1$ and $r=-1$. Discuss the convergence or divergence of the series in these cases. Do you really need to use a computer in part $c$ or even in part $b$ ? Explain. d. When $r$ is negative in a geometric series we get an example of an alternating series. Explain why this is an aptly chosen description. Give examples of two alternating geometric series, one convergent and one divergent. e. Investigate the behavior of the geometric series for various values of the ratio $r$. Write a clear and complete statement summarizing your results.

A friendly warm-up. Consider the geometric series $\sum_{k=0}^{\infty} r^k$, where the real number $r$ gives the ratio between successive terms. Note that the series starts with $k=0$ rather than $k=1$.
a. Let $r=\frac{4}{5}$ and use the computer to investigate the partial sums of the series $\sum_{k=0}^{\infty}\left(\frac{4}{5}\right)^k$. As you evaluate partial sums with more and more terms, what appears to be the limiting value? Confirm that this value for the sum of the infinite series agrees with the well-known formula for the sum of a geometric series whose ratio $r$ has absolute value less than 1 .
b. Let $r=1.01$ and use the computer to study the partial sums. What do you conclude about the convergence or divergence of the geometric series in this case? Explain.
c. Now let $r=1$ and $r=-1$. Discuss the convergence or divergence of the series in these cases. Do you really need to use a computer in part $c$ or even in part $b$ ? Explain.
d. When $r$ is negative in a geometric series we get an example of an alternating series. Explain why this is an aptly chosen description. Give examples of two alternating geometric series, one convergent and one divergent.
e. Investigate the behavior of the geometric series for various values of the ratio $r$. Write a clear and complete statement summarizing your results.

Key Concepts

Alternating Series

An alternating series is characterized by successive terms that change sign, typically due to a negative common ratio in geometric series. The alternating nature can lead to different convergence behavior than non-alternating series, and it is often analyzed with tests specifically designed to handle sign changes.

Partial Sums

Partial sums involve summing only the first n terms of a series, which provides an approximation of the total sum. Studying these sums is crucial for understanding how quickly a series converges to its limiting value or whether it diverges.

Sum Formula for Geometric Series

When the absolute value of the common ratio is less than one, the infinite geometric series converges and its sum is given by the formula S = a / (1 - r), where a is the first term. This formula allows for the direct computation of the limit without needing to sum all infinitely many terms.

Convergence Criteria

Convergence in an infinite series means that as more terms are added, the sequence of partial sums approaches a fixed finite value. For geometric series, the key criterion for convergence is that the absolute value of the common ratio must be less than one; if it is equal to or greater than one, the series diverges.

Geometric Series

A geometric series is a sum of successive terms where each term is obtained by multiplying the previous term by a constant ratio. This structure is fundamental in both pure and applied mathematics, illustrating concepts such as exponential growth and decay.

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