The following series are geometric series or a sum of two geometric series. determine whether each series converges or not. for the series which converge, enter the sum of the series. for the series which diverges enter "div" (without quotes). (a) ∑n=1∞10n9n= , (b) ∑n=2∞13n= , (c) ∑n=0∞3n52n+1= , (d) ∑n=5∞9n10n= , (e) ∑n=1∞7n7n+4= , (f) ∑n=1∞9n+3n10n=

a. Does not converge

b. Converges. And the sum would be 0.5

c. Converges. And the sum would be 2.411

Step-by-step explanation:

Remember that a series of the type

Converges when  

a.

Therefore it does not converge.

b.  

Therefore it converges. And the sum would be

c.

Therefore it converges. And the sum would be

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.