Ratio Test: - Consider a series where the terms can be expressed as a ratio of two numbers. - The test says that the series converges if the absolute value...
Ratio Test:
Consider a series where the terms can be expressed as a ratio of two numbers.
The test says that the series converges if the absolute value of the ratio is less than 1.
If the absolute value of the ratio is greater than 1, the series diverges.
For example, consider the series ( \sum_{n=1}^\infty \frac{(-1)^n}{n!} ). This series converges because (| \frac{(-1)^n}{n!} | = \frac{1}{n} < 1 ).
Root Test:
Consider a series where the terms can be expressed as a root of a number.
The test says that the series converges if the real part of the complex conjugate of the root is less than 1.
If the real part of the complex conjugate of the root is greater than 1, the series diverges.
For example, consider the series ( \sum_{n=1}^\infty \left( \frac{n}{n+1} \right)^n ). This series converges because (\frac{n}{n+1} > 1) for all (n \ge 1)