Test Convergence

Convergence of a sequence:

A sequence is defined as a convergent sequence if it converges to a real number.

Divergence of a sequence:

A sequence is defined as a divergent sequence if it converges to 8, which is not a real number.

Convergent series:

A series is called as a convergent series if an infinite series for which the sequence of partial sums converges or meets.

Divergent series:

A series is called as a divergent series if it does not converge.

Tests for Convergence:

These are the procedures to test the convergence, conditional convergence, whole convergence or absolute convergence etc. The various tests involved are:

• Limit of the summand
• Ratio test
• Integral test
• Limit Comparison test
• Cauchy condensation test
• Abel's test

Divergence Tests:

If the limit of a sequence is not zero or does not exist then the sum is said to be divergent.

Consider an example:∑ 1 + n/n where n = 1 to ∞

This does not converge as the limit is 1 as n tends to infinity. One has to understand that the assumption only goes one way, ie., if the limit = 0, it may not converge.

Consider another eg:

∑ 1/n where n = 1 to ∞

This might have a limit = 0, but the sum does not converge.

How to test the convergence for a Power series:

• Inorder to find the interval in which the series converges absolutely, Ratio test or nth Root test can be used.
• If the absolute convergence interval is finite and the test for convergence/divergence exists at the each of the 2 endpoints, comparison test, integral test or the alternating series theorem can be used.

Summary:

The root test is powerful than the ratio test since the required condition is weaker. But, whenever the ratio test determines the convergence or divergence, the root test also does, but vice versa is not possible.