Which statement best describes convergent infinite series as used in the discussion?

Which statement best describes convergent infinite series as used in the discussion?

• A. Infinitely many terms always sum to infinity.
• B. Infinitely many terms can sum to a finite limit.
• C. The sum of infinite terms cannot be determined.
• D. Only finite sums are meaningful in practice.

Answer: (B)

The idea being tested is whether adding up infinitely many terms can settle to a finite total. You look at the partial sums S_n = a1 + a2 + ... + an and ask if, as n grows larger, S_n approaches some finite value. If it does, the series is convergent and the infinite sum equals that limit. A classic example is 1/2 + 1/4 + 1/8 + ..., whose partial sums get closer and closer to 1. This shows that infinitely many terms can indeed sum to a finite limit. The other statements don’t fit because a convergent series does not blow up to infinity; it has a finite sum. And while not every infinite series has an easily found closed form, its sum can still be defined as the finite limit of the partial sums. Infinite series are also used in practice precisely because many converge to useful finite values.

The idea being tested is whether adding up infinitely many terms can settle to a finite total. You look at the partial sums S_n = a1 + a2 + ... + an and ask if, as n grows larger, S_n approaches some finite value. If it does, the series is convergent and the infinite sum equals that limit. A classic example is 1/2 + 1/4 + 1/8 + ..., whose partial sums get closer and closer to 1. This shows that infinitely many terms can indeed sum to a finite limit.

The other statements don’t fit because a convergent series does not blow up to infinity; it has a finite sum. And while not every infinite series has an easily found closed form, its sum can still be defined as the finite limit of the partial sums. Infinite series are also used in practice precisely because many converge to useful finite values.