Question

If a series ∑n=0[infinity]an converges, then the series ∑n=0[infinity](50an) also converges.

a true
b false

Answer 1

If a series ∑n=0∞ an converges, then the series ∑n=0∞ (50an) also converges.

Step-by-step explanation:

The statement is true:

If a series ∑n=0∞ an converges, then the series ∑n=0∞ (50an) also converges.

To prove this, we can use the fact that if a series converges, then the sequence of its terms must approach zero. Let's assume that ∑n=0∞ an converges. This means that limn→∞ an = 0.

Now let's consider the series ∑n=0∞ (50an). Using the limit properties, we can rewrite it as 50∑n=0∞ an. Since 50 is a constant, it does not affect the convergence of the series. Therefore, if limn→∞ an = 0, then limn→∞ (50an) = 50 * 0 = 0.

This shows that ∑n=0∞ (50an) also converges, proving the statement true.