Question 09 (a)
Long Problem. Show your work. No credit will be given for the answer without the correct accompanying work.
Show that for .
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.
Consider geometric series.
How does the derivative affect the series?
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
We know that when a geometric series converges,
A geometric series converges when which is what we have in the problem. If we differentiate both sides:
To get the extra factor of , multiply both sides by :
This multiplication is okay to do since the sum changes with and not . This same logic is why we were not allowed to start by multiplying both sides by .
We have shown the series relationship holds.