Ratio Test for Convergence of an Infinite Series
Key Questions

It is not always clearcut, but if a series contains exponential functions or/and factorials, then Ratio Test is probably a good way to go.
I hope that this was helpful.

By Ratio Test, the posted series converges absolutely.
By Ratio Test:
#lim_{n to infty}a_{n+1}/a_n=lim_{n to infty}{(10)^{n+1}}/{4^{2n+3}(n+2)}cdot{4^{2n+1}(n+1)}/{(10)^n}# By canceling out common factors:
#=lim_{n to infty}{10(n+1)}/{4^2(n+2)}# since
#{10}/4^2=5/8# , we have:#=5/8lim_{n to infty}(n+1)/(n+2)# by dividing the numerator and the denominator by
#n# ,#=5/8 lim_{n to infty}{1+1/n}/{1+2/n}=5/8cdot 1=5/8<1# Hence,
#sum_{n=1}^{infty}{(10)^n}/{4^{2n+1}(n+1)}# is absolutely convergent.
Questions
Tests of Convergence / Divergence

Geometric Series

Nth Term Test for Divergence of an Infinite Series

Direct Comparison Test for Convergence of an Infinite Series

Ratio Test for Convergence of an Infinite Series

Integral Test for Convergence of an Infinite Series

Limit Comparison Test for Convergence of an Infinite Series

Alternating Series Test (Leibniz's Theorem) for Convergence of an Infinite Series

Infinite Sequences

Root Test for for Convergence of an Infinite Series

Infinite Series

Strategies to Test an Infinite Series for Convergence

Harmonic Series

Indeterminate Forms and de L'hospital's Rule

Partial Sums of Infinite Series