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Comparison Tests

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Comparison Tests
Consider the two series ∑n=1∞an\sum\limits^\infty_{n=1}a_nn=1∑∞​an​ and ∑n=1∞bn\sum\limits^\infty_{n=1}b_nn=1∑∞​bn​ with non-negative terms. 
Series Comparison Test
If an≤bna_n \le b_nan​≤bn​ after a certain point, then
if ∑n=1∞bn\sum\limits^\infty_{n=1}b_nn=1∑∞​bn​ converges, then ∑n=1∞an\sum\limits^\infty_{n=1}a_nn=1∑∞​an​ converges
if ∑n=1∞an\sum\limits^\infty_{n=1}a_nn=1∑∞​an​ diverges, then ∑n=1∞bn\sum\limits^\infty_{n=1}b_nn=1∑∞​bn​ diverges.
Limit Comparison Test
Assume that all the terms ana_nan​ and bnb_nbn​ are positive after a certain point.
If lim⁡n→∞anbn=k>0\lim\limits_{n\rightarrow\infty}\frac{a_n}{b_n}=k>0n→∞lim​bn​an​​=k>0, then ether ∑n=1∞an\sum\limits^\infty_{n=1}a_nn=1∑∞​an​  and ∑n=1∞bn\sum\limits^\infty_{n=1}b_nn=1∑∞​bn​ both converge or both diverge
If lim⁡n→∞anbn=0\lim\limits_{n\rightarrow\infty}\frac{a_n}{b_n}=0n→∞lim​bn​an​​=0, then if ∑n=1∞bn\sum\limits^\infty_{n=1}b_nn=1∑∞​bn​ converges, then ∑n=1∞an\sum\limits^\infty_{n=1}a_nn=1∑∞​an​ must also converge
If lim⁡n→∞anbn=∞\lim\limits_{n\rightarrow\infty}\frac{a_n}{b_n}=\inftyn→∞lim​bn​an​​=∞ , then if ∑n=1∞bn\sum\limits^\infty_{n=1}b_nn=1∑∞​bn​ diverges, then ∑n=1∞an\sum\limits^\infty_{n=1}a_nn=1∑∞​an​ must also diverge.
Identifying Clues
The series looks like a p-series, geometric, or harmonic
They give you another series in the question that looks similar to the series in question, and tell you whether that series converges or diverges
Some useful tricks:
ln⁡n>1\ln n>1lnn>1 for n≥3n\ge3n≥3
ln⁡n<1\ln n<1lnn<1 for n<3n<3n<3
1+big term<1+small term\frac{1}{+big\ term}<\frac{1}{+small\ term}+big term1​<+small term1​ and 1−big term>1−small term\frac{1}{-big\ term}>\frac{1}{-small\ term}−big term1​>−small term1​
−1≤sin⁡n, cos⁡n ≤1-1\le\sin n,\ \cos n\ \le1−1≤sinn, cosn ≤1
−π2<arctan⁡n<π2-\frac{\pi}{2}<\arctan n<\frac{\pi}{2}−2π​<arctann<2π​

Practice: Limit Comparison Test
Determine whether the series ∑n=1∞2n+n{\displaystyle \sum_{n=1}^\infty}\frac{2}{n+\sqrt{n}}n=1∑∞​n+n​2​ converges or diverges.

It converges

It diverges

Cannot be determined

I don't know

Practice: Limit Comparison Test
Determine if the series ∑n=1∞n+n3n2+n3\sum\limits^\infty_{n=1}\frac{\sqrt{n}+\sqrt[3]{n}}{n^2+n^3}n=1∑∞​n2+n3n​+3n​​ converges or diverges.

It converges

It diverges

Cannot be determined

I don't know

Practice Question
Determine whether the series ∑n=2∞−2n2−1{\displaystyle \sum_{n=2}^\infty}-\frac{2}{n^2-1}n=2∑∞​−n2−12​ converges or diverges. 
If it converges, find the value it converges to.

Practice: Series Comparison Test
Determine whether the series ∑n=1∞en(3n)2\sum\limits^\infty_{n=1}\frac{\sqrt[n]{e}}{(3n)^2}n=1∑∞​(3n)2ne​​ converges or diverges.

It converges

It diverges

Cannot be determined

I don't know

Practice: Series Comparison Test
Determine if the series ∑n=1∞sin⁡nn3{\displaystyle \sum_{n=1}^\infty}\frac{\sin n}{n^3}n=1∑∞​n3sinn​ converges or diverges.

It converges

It diverges

Cannot be determined

I don't know

Practice Question
Determine whether the series ∑n=1∞(5n+1−5n+2){\displaystyle \sum_{n=1}^\infty}\left(\frac{5}{n+1}-\frac{5}{n+2}\right)n=1∑∞​(n+15​−n+25​) converges or diverges. 
If it converges, find the value it converges to.