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Using the steps below, show that the following sequence converges:

[itex]1+\frac{1}{2}-\frac{2}{3}+\frac{1}{4}+\frac{1}{5}-\frac{2}{6}+\frac{1}{7}+\frac{1}{8}-\frac{2}{9}+\frac{1}{10}+\frac{1}{11}-\frac{2}{12}++-++-...[/itex]

i. Consider the subsequence (s2,s3,s5,s6,s8,s9,...) of the sequence of partial

sums. Show that this is the sequence of partial sums of a related convergent

series.

ii. Show that the original series is also convergent.

I have tried and showed part ii successfully, by considering

[itex]\frac{1}{2}-\frac{2}{3}+\frac{1}{5}-\frac{2}{6}+\frac{1}{8}-\frac{2}{9}+\frac{1}{11}-\frac{2}{12}+...[/itex]

[itex]=(\frac{1}{2}+\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+...)-(\frac{2}{3}+\frac{2}{6}+\frac{2}{9}+\frac{2}{12}+...)[/itex]

[itex]=\sum\limits_{k=1}^\infty \frac{1}{3k-1}-\sum\limits_{k=1}^\infty \frac{2}{3k}[/itex]

and considering

[itex]1+\frac{1}{4}+\frac{1}{7}+\frac{1}{10}+...[/itex]

[itex]=\sum\limits_{k=1}^\infty \frac{1}{3k-2} [/itex]

So the sequence =[itex] \sum\limits_{k=1}^\infty \frac{9k-4}{3k(3k-2)(3k-1))} [/itex]

is convergent by comparison test

However I am wondering what part (i) is asking. I think (s2,s3,s5,s6,s8,s9,...) is divergent. How can I relate to a convergent series?

[itex]1+\frac{1}{2}-\frac{2}{3}+\frac{1}{4}+\frac{1}{5}-\frac{2}{6}+\frac{1}{7}+\frac{1}{8}-\frac{2}{9}+\frac{1}{10}+\frac{1}{11}-\frac{2}{12}++-++-...[/itex]

i. Consider the subsequence (s2,s3,s5,s6,s8,s9,...) of the sequence of partial

sums. Show that this is the sequence of partial sums of a related convergent

series.

ii. Show that the original series is also convergent.

I have tried and showed part ii successfully, by considering

[itex]\frac{1}{2}-\frac{2}{3}+\frac{1}{5}-\frac{2}{6}+\frac{1}{8}-\frac{2}{9}+\frac{1}{11}-\frac{2}{12}+...[/itex]

[itex]=(\frac{1}{2}+\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+...)-(\frac{2}{3}+\frac{2}{6}+\frac{2}{9}+\frac{2}{12}+...)[/itex]

[itex]=\sum\limits_{k=1}^\infty \frac{1}{3k-1}-\sum\limits_{k=1}^\infty \frac{2}{3k}[/itex]

and considering

[itex]1+\frac{1}{4}+\frac{1}{7}+\frac{1}{10}+...[/itex]

[itex]=\sum\limits_{k=1}^\infty \frac{1}{3k-2} [/itex]

So the sequence =[itex] \sum\limits_{k=1}^\infty \frac{9k-4}{3k(3k-2)(3k-1))} [/itex]

is convergent by comparison test

However I am wondering what part (i) is asking. I think (s2,s3,s5,s6,s8,s9,...) is divergent. How can I relate to a convergent series?

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