Question

Verify that the infinite series diverges. $$ \sum_{n=0}^{\infty} 2(-1.03)^{n} $$

Short answer

The infinite geometric series \(\sum_{n=0}^{\infty} 2(-1.03)^{n}\) diverges since the absolute value of the ratio (1.03) is greater than 1.

Step-by-step solution

Identify the Series Type

The series provided can be written in the form of a geometric series, which is \(\sum a*r^{n}\). Here, \(a = 2\) and \(r = -1.03\).

Evaluate the Ratio

The constant ratio for this geometric series is \(r = -1.03\).

Apply the Convergence Condition

A geometric series \(\sum ar^{n}\) is convergent if \(-1 < r < 1\) and divergent otherwise. Since the \(r\) value is greater than 1 or less than -1, the series is divergent.