Direct Comparison Test for Convergence of an Infinite Series
Key Questions

By Comparison Test, we can conclude that the series
#sum_{n=1}^infty{9^n}/{3+10^n}# converges.Let us look at some details.
For all#n geq 1# ,
#9^n/{3+10^n} leq 9^n/10^n=(9/10)^n# By Geometric Series Test,
#sum_{n=1}^infty(9/10)^n# converges since#r=9/10<1# Hence, by Comparison Test,
#sum_{n=1}^infty{9^n}/{3+10^n}# also converges. 
If an improper integral diverges, then we probably do not want to wast time trying to find its value.
Let us assume that we already know:
#int_1^infty1/x dx=infty# Let us look examine this uglier improper integral.
#int_1^infty{4x^2+5x+8}/{3x^3x1} dx# By making the numerator smaller and the denominator bigger,
#{4x^2+5x+8}/{3x^3x1} ge {3x^2}/{3x^3}=1/x# By Comparison Test, we may conclude that
#int_1^infty{4x^2+5x+8}/{3x^3x1} dx# diverges.(Intuitively, if the smaller integral diverges, then the larger one has no chance to converge.)
I hope that this was helpful.

If you are trying determine the conergence of
#sum{a_n}# , then you can compare with#sum b_n# whose convergence is known.If
#0 leq a_n leq b_n# and#sum b_n# converges, then#sum a_n# also converges.
If#a_n geq b_n geq 0# and#sum b_n# diverges, then#sum a_n# also diverges.This test is very intuitive since all it is saying is that if the larger series comverges, then the smaller series also converges, and if the smaller series diverges, then the larger series diverges.
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