Problem 83 - Essential Calculus

Question

(a) You delete a finite number of terms from a divergent series. Will the new series still diverge? Explain your reasoning. (b) You add a finite number of terms to a convergent series. Will the new series still converge? Explain your reasoning.

Step-by-Step Solution

Answer

Yes to both. Since we are adding or removing only a finite number of terms in both scenarios, the core properties of the series (convergence or divergence) remain the same.

Step 1: Understand the Principles of Convergence and Divergence

A series is said to converge if the sum of its terms tends towards a specific, finite number as the number of terms approaches infinity. Conversely, a series is said to diverge if the sum of its terms tends towards infinity, or does not tend towards a specific, finite number as the number of terms approaches infinity.

Step 2: Apply the Principle of Convergence

Consider part (b) of the exercise. Adding a finite number of terms to a convergent series will still result in a convergent series. The sum of a finite number of terms will always be finite. Thus, adding a finite amount to the sum of a convergent series will not change its status to divergent, since the overall sum still approaches a specific, finite number when the number of terms approaches infinity.

Step 3: Apply the Principle of Divergence

For part (a), deleting a finite number of terms from a divergent series does not change the property of divergence. This is because even after removing a finite number of terms, as the number of terms of the series tends to infinity, the sum of the series will still be infinity or it still will not approach a specific, finite number. Hence, the series remains divergent after removing a finite number of terms.