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Answer:
Required solution gives series (a) divergent, (b) convergent, (c) divergent.
Step-by-step explanation:
(a) Given,
To applying limit comparison test, let and
. Then,
Because of the existance of limit and the series is divergent since
where
, given series is divergent.
(b) Given,
Again to apply limit comparison test let and
we get,
Since is convergent, by comparison test, given series is convergent.
(c) Given,
. Now applying Cauchy Root test on last two series, we will get,
- \lim_{n\to \infty}|(\frac{5}{6})^n|^{\frac{1}{n}}=\frac{5}{6}=L_1
- \lim_{n\to \infty}|(\frac{1}{3})^n|^{\frac{1}{n}}=\frac{1}{3}=L_2
Therefore,
Hence by Cauchy root test given series is divergent.