Summary

Good to Know

Any Series diverges if:

  • $\lim\limits_{n\rightarrow\infty}s_n = c \neq 0$

If $n$ is close enough to infinity, we’ll be constantly adding a constant $c$

Regardless of it’s value, if it’s not 0, it’ll tend to $+\infty$ or $-\infty$

Alternating Series Test

Series Converges if:

  • $\lim\limits_{n\rightarrow\infty}s_n = 0$
  • $\lvert a_{n+1}\rvert \leq \lvert a_{n}\rvert$
    • It can be $\leq$ because $\lim\limits_{n\rightarrow\infty}s_n = 0$ guarantees it’ll end up at $0$

Integral Test

If it’s possible to map $a_n = f(n)$

If $f(n)$ is:

  • Continuous
  • Decreasing
  • Positive, hence $a_n$ must be all positive

If $N$ is some positive integer

$\int_{N}^{\infty} f(n)$ converges then $\sum_{n=N}^{\infty}a_n$ converges

$\int_{N}^{\infty} f(n)$ diverges then $\sum_{n=N}^{\infty}a_n$ diverges

Comparison Test

If $a_n > 0$ and $b_n > 0$

If $a_n \leq b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges

If $a_n \geq b_n$ and $\sum b_n$ diverges, then $\sum a_n$ diverges

Limit Comparison Test

If $a_n > 0$ and $b_n > 0$

If $N$ is some positive integer

$\lim_{n\rightarrow\infty}(a_n/b_n)$ = Test Result

Result a;b
$> 0$ Converge;Converge OR Diverge;Diverge
$= 0$ Converge;Converge
$\infty$ Diverge;Diverge
Other No Conclusion

Absolute/Conditional Convergence

If $\sum a_n$ converges, it’s Convergent

If $\sum\lvert a_n\rvert$ converges, it’s Absolutely Convergent

If a series is Absolutely Convergent, it’s Convergent.

If a series is Convergent but not Absolutely Convergent, it’s Conditionally Convergent

P-Series

\[\sum_{n=0}^{\infty}1/(p^n)\]
p  
$< 1$ Converge
$\geq 1$ Diverge

Ratio Test

$\lim\limits_{n\rightarrow\infty}\lvert a_{n+1}/a_n\rvert$ = Test Result

Result  
$< 1$ Absolute Converge
$> 1$ Diverge
$= 1$ No Conclusion

Root Test

$\lim\limits_{n\rightarrow\infty}(a_n)^{1/n}$ = Test Result

Result  
$< 1$ Absolute Converge
$> 1$ Diverge
$= 1$ No Conclusion