Science:Math Exam Resources/Courses/MATH101 B/April 2025/Question 08 (c)

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MATH101 B April 2025

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q2 (a)  •  Q2 (b)  •  Q2 (c)  •  Q3 (a)  •  Q3 (b)  •  Q4  •  Q5  •  Q6  •  Q7 (a)  •  Q7 (b)  •  Q8 (a)  •  Q8 (b)  •  Q8 (c)  •  Q9  •  Q10  •  

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  •  April 2024  •  April 2025  •  

Question 08 (c)
In this question, $a$ is a real number. Find, with justification, all values of $a$ for which the series $$\begin{array}{r}\sum _{n=1}^{\mathrm{\infty }}{e}^n{e}^{-an}.\end{array}$$ converges. Evaluate the series (in terms of $a$ ) for all values of $a$ for which it converges.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Rewrite $e^n{e}^{-an}=e^{(1-a)n}$ and view the series as a geometric series with ratio $r=e^{1-a}$. For which values of $r$ does a geometric series converge?

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

We can rewrite the series as follows: $$\sum _{n=1}^{\mathrm{\infty }}{e}^n{e}^{-an}=\sum _{n=1}^{\mathrm{\infty }}{e}^{(1-a)n}.$$ Let $r=e^{1-a}>0$. Then the series becomes $$\sum _{n=1}^{\mathrm{\infty }}{r}^n,$$ which is a geometric series. A geometric series converges if and only if $|r|<1$. Since $r>0$, this is equivalent to $$e^{1-a}<1.$$ Taking logarithms, $$1-a<0⟺a>1.$$ Thus, the series converges if and only if $a>1$. For $a>1$, we can evaluate the sum: $$\sum _{n=1}^{\mathrm{\infty }}{r}^n=\frac{r}{1-r}.$$ Substituting $r=e^{1-a}$, we obtain $$\sum _{n=1}^{\mathrm{\infty }}{e}^n{e}^{-an}=\frac{e^{1-a}}{1-e^{1-a}}=\frac{1}{e^{a-1}-1}.$$ Therefore, the series converges if and only if $a>1$, and in that case $$\sum _{n=1}^{\mathrm{\infty }}{e}^n{e}^{-an}=\frac{1}{e^{a-1}-1}.$$

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