Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\sum_{n=1}^{\infty} a_{n}=L,\) then \(\sum_{n=0}^{\infty} a_{n}=L+a_{0}\).

Short answer

The statement is true. By definition of an infinite series starting from n=0, the sum would naturally be the sum of the first term and the sum of the rest of the series.

Step-by-step solution

Understanding the problem

In the given statement, it's implied that adding the first term of the series (when n is 0), to the sum from n=1 to infinite, will give the sum from n=0 to infinite. This assertion seems logical, because adding a term to a sum would normally change the sum by that amount.

Evaluating the truthfulness

The given statement is actually true. If we have a series starting from n=0, then the first term will be given when n=0, i.e., \( a_{0} \), and rest of the series would be from n=1 to infinity, i.e., \( \sum_{n=1}^{\infty} a_{n} \). So, it's natural that the sum from n=0 to infinity, \( \sum_{n=0}^{\infty} a_{n} \), would be the sum of these, i.e., \( L+a_{0} \).