Shelbi wants to run one lap around the track as fast as her friend Annabelle, who can run one lap in 1 minute and 10 seconds. Sh
2 answers:
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Best guess for the function is
By the ratio test, the series converges for
When ,
is a convergent
-series.
When ,
is a convergent alternating series.
So, the interval of convergence for is the <em>closed</em> interval
.
The derivative of is the series
which also converges for by the ratio test:
When ,
becomes the divergent harmonic series.
When ,
is a convergent alternating series.
The interval of convergence for is then the <em>closed-open</em> interval
.
Differentiating once more gives the series
The first series is geometric and converges for , endpoints not included.
The second series is , which we know converges for
.
Putting these intervals together, we see that converges only on the <em>open</em> interval
.