43. Additional Convergence Tests

Dated: 30-10-2024

The Comparison Test

Let us say there are two series¹ \(\sum a_i\) and \(\sum b_i\) with non negative terms such that

\[a_1 \le b_1, a_2 \le b_2, \ldots a_i \le b_i\]

Then 1. If \(\sum b_i\) converges then \(\sum a_i\) also converges. 2. If \(\sum a_i\) diverges then \(\sum b_i\) also diverges.

The Ratio Test

Let \(\sum u_i\) be a series¹ and \(p = \lim_{i \to \infty} \frac {u_{i + 1}}{u_i}\) then if

\(p < 1\) then series¹ converges.
\(p > 1\) then series¹ diverges.
\(p = 1\) then series¹ may either converge or diverge.

The Root Test

Let \(\sum u_i\) be a series¹ and \(p = \lim_{i \to \infty} (u_i)^{\frac 1 i}\) then if

\(p < 1\) then series¹ converges.
\(p > 1\) then series¹ diverges.
\(p = 1\) then series¹ may either converge or diverge.

Informal Principal

The constant terms in the denominator can usually be deleted without affecting the convergence and divergence of the series.¹
If a polynomial² in \(i\) (or whatever index used for sum³) appears as a factor in numerator or denominator then all the \(i\) terms except for the the one with the highest power, can usually be deleted without affecting the convergence and divergence of the series.¹

Examples

The following 2 behaves alike

\[\sum_{i=1}^{\infty}\frac{1}{2^{i}+1}\]

\[\sum_{i=1}^{\infty}\frac{1}{2^{i}}\]

The following 2 behaves alike

\[\sum_{i=1}^{\infty}\frac{1}{\sqrt{i^{3}+2i}}\]

\[\sum_{i=1}^{\infty}\frac{1}{\sqrt{i^{3}}}\]

The Limit Comparison Test

Let us say there are two series¹ \(\sum a_i\) and \(\sum b_i\) with positive terms such that \(p = \lim_{i \to +\infty}\frac{a_i}{b_i}\).
Then if \(p > 0\) and is finite then both series¹ either converge or diverge.