Science:Math Exam Resources/Courses/MATH103/April 2014/Question 01 (e) ii

MATH103 April 2014

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Question 01 (e) ii
$(a_{k})_{k=1}^{\infty }=\left(1,0,{\frac {1}{2}},0,0,{\frac {1}{3}},0,0,0,{\frac {1}{4}},0,0,0,0,{\frac {1}{5}},\ldots \right)$ and its associated series $\sum _{k=1}^{\infty }a_{k}$. True or false: the series $\sum _{k=1}^{\infty }a_{k}$ diverges.

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
The series $\sum _{k=1}^{\infty }a_{k}$ is the same as a very well-known series.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The series is $1+0+{\frac {1}{2}}+0+0+{\frac {1}{3}}+\cdots =1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots ,$ which is just the harmonic series. But the harmonic series diverges (say by using the p-series test if need be)! Note the difference between a sequence and a series. The statement is true.

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