10.1.7Take it to the limit.

Harry is thinking about the infinite series $T = \displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n + 0.5 }$. Harry assumes that $T$ behaves in a similar way to the harmonic series $S = \displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$, which, according to the $p$-Series Test, diverges. Harry quickly concluded that $T$ must diverge as well.

His teammate Henrietta challenged him, “Not so fast, Harry! You cannot draw a conclusion without comparing the corresponding terms of $S$ and $T$.”

1. List and compare the corresponding terms of $S$ and $T$. Do these series meet the criteria of the Comparison Test? Explain.

2. Explain why using $W = \displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } }$ to determine that $U = \displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( n - 0.5 ) ^ { 2 } }$ converges is also a violation of the Comparison Test.

3. Harry is heartbroken because the Comparison Test failed to prove the divergence of $T$ or the convergence of $U$. Does this mean that $T$ converges and $U$ diverges? Explain your thoughts. 

Even without the Comparison Test to back him up, Harry still believes that $S$ should behave like $T$ and that $U$ should behave like $W$. “After all,” he argues, “they are just horizontal shifts of each other! I don’t think that a shift will affect the convergence of an infinite series!”

Once again, consider the series $S = \displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$ and $T = \displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n + 0.5 }$.

Let $a_n$ be the $n^\text{th}$ term of $S$ and $b_n$ be the $n^\text{th}$ term of $T$.

1. What are $a_{10}$ and $b_{10}$? $a_{100}$ and $b_{100}$? Compare the ratio $\frac { a _ { 10 } } { b _ { 10 } }$ with $\frac{a_{100}}{b_{100}}$.

2. What is $\lim\limits _ { n \rightarrow \infty } a _ { n }$? What is $\lim\limits _ { n \rightarrow \infty } b _ { n }$? How about the ratio between these two limits, $\lim\limits _ { n \rightarrow \infty } \frac { a _ { n } } { b _ { n } }$? Can you use this result to support your claim regarding the convergence of $T$?

Harry is at it again. While investigating the series $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { e ^ { n - 1 } }$, he suspects that it converges. He attempts to demonstrate his suspicions by using a limit to compare it to the series $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } }$, which he knows converges by the $p$-Series Test.

1. Demonstrate what Harry is thinking. Evaluate $\lim\limits _ { n \rightarrow \infty } \frac { a _ { n } } { b _ { n } }$, where $a_n$ represents the argument of one of the series and $b_n$ represents the argument of the other. What happens? Does this result demonstrate that $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { e ^ { n - 1 } }$ must behave like the convergent series $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } }$?

2. Harry is not sure how to interpret the result of his limit. He notices that Henrietta wrote her limit differently. She used $\lim\limits _ { n \rightarrow \infty } \frac { b _ { n } } { a _ { n } }$; yet she is also uncertain. What was Henrietta’s result and why was she uncertain?

3. Henrietta is frustrated. She wonders if this result means that $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { e ^ { n - 1 } }$ diverges, so she compares it to the harmonic series $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$. Use $\lim\limits _ { n \rightarrow \infty } \frac { a _ { n } } { b _ { n } }$ to test her strategy.

4. Meanwhile, Harry still believes that $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { e ^ { n - 1 } }$ converges. He decides to compare it to a similarly behaved series, $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { e ^ { n } }$, which he remembers converges after doing problem 10-69. Evaluate this limit. Does this limit indicate that $\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { e ^ { n - 1 } }$ converges?  

5. Write a summary statement for Harry and Henrietta about how to make a good choice when selecting a series in which to compare an unknown series.

The Limit Comparison Test

Copy and complete the statement below to write a conjecture describing how two similarly-behaved series can be used to determine if they both converge or both diverge.

Use an appropriate test to determine if each of the following series converges or diverges.

1. Integral Test
2. Comparison Test
3. Limit Comparison Test

Revisit the series in problem 10-2. For which series could you have used the Comparison or Limit Comparison Test to determine convergence? For each, use either test to determine if the series converges.

By now you are comfortable with the graphical implications of the first and second derivatives of a function. $f^\prime$ gives you the slope of the tangent line at a point while $f^{\prime\prime}$ tells you the concavity of the graph at a point. There is nothing to stop us from finding the third and fourth (and beyond!) derivatives, although there are not any significant graphical characteristics associated with higher derivatives. $f^{\prime\prime\prime}$ is simply the rate of change of $f^{\prime\prime}$ and so on. Homework Help ✎

1. What is $f^{(4)}(x)$ if $f(x) = x^8$?

2. If $f(x) = e^{2x}$, write an expression for the $n^{\text{th}}$ derivative, $f^{(n)}(x)$.

Describe the curve whose parametric equations are $x(t) = a \cos(t) + h$ and $y(t) = b \sin(t) + k$, for $0 ≤ t ≤ 2π$. Be explicit. 10-85 HW eTool (Desmos). Homework Help ✎

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy for each part. Evaluate each integral and briefly describe your method. Homework Help ✎

Examine the following series. Use tests that you have learned so far to determine if each series converges or diverges. Name the tests that you use. Homework Help ✎

Calculate the length of the curve $y=\frac { 1 } { 2 }x + 1$ from $x = 0$ to $x = 5$. Homework Help ✎