Science:Math Exam Resources/Courses/MATH103/April 2014/Question 11

MATH103 April 2014

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[hide] Question 11
'This is a difficult question and will be marked very strictly. You should try this ONLY AFTER you finish all other problems. Suppose a sequence $(b_{n})_{n\geq 0}$ satisfies the recursive relation $b_{n+1}=\beta {\frac {b_{n}}{1+b_{n}}},$ where $\beta$ is a given positive constant. Suppose $b_{0}=1$ . Find all positive constants $\beta$ for which the series $\sum _{n=0}^{\infty }b_{n}$ converges. Justify your answer.

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.

[show] Hint 1
Do not use the ratio test (you will get a ratio of $1$ , so the test is indeterminate).

[show] Hint 2
Use the divergence test.

[show] Hint 3
When is $0$ a stable fixed point? When is it the only stable fixed point?

[show] Hint 4
If $0$ is a fixed point of $F$ for which $\|F'(0)\|=1$ , then we can’t tell if it is stable or unstable. Make sure to check what happens in these special cases.

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[show] Solution
By the divergence test, it is necessary (though perhaps not sufficient) that $b_{n}\to 0$ . Thus, the map $F(x)=\beta {\frac {x}{1+x}}$ must at least have a stable fixed point at $0$ . We thus begin by find the fixed points of $F$ . Solving $F(x)=x$ yields $x=0$ or $x=\beta -1$ . To check stability, note that $F'(x)={\frac {\beta }{(1+x)^{2}}}$ . Since $F'(0)=\beta >0$ , we require $\beta \leq 1$ for $0$ to be stable; we will deal with the indeterminate case $\beta =1$ later. If $\beta <1$ , we use the ratio test to see that $\lim _{n\to \infty }\left\|{\frac {b_{n+1}}{b_{n}}}\right\|=\lim _{n\to \infty }\left\|{\frac {\beta }{1+b_{n}}}\right\|=\beta$ . We conclude by the ratio test that the series converges for $\beta <1$ and diverges for $\beta >1$ . In the case $\beta =1$ , we have $b_{n+1}={\frac {b_{n}}{1+b_{n}}}$ . With $b_{0}=1$ , looking at the first few terms of the resulting sequence, we see that $b_{1}=1/2,b_{2}=1/3,b_{3}=1/4$ and so on. We thus see that $b_{n}=1/(n+1)$ . But then the series is the harmonic series, which diverges. Thus, the series converges for $0<\beta <1$ .