Question

Consider two series:

[ Series A: ∑ₙ=1[infinity]1/π n; Series B: ∑ₙ=0[infinity]1/(e+π)² n ]
Which of these series converges?

Answer 1

Of the two series presented, Series A diverges as it is a p-series with p=1, and Series B converges as it is a geometric series with a common ratio less than 1.

Step-by-step explanation:

The given question involves two infinite series and whether they converge or diverge. Looking at Series A, it is given by ∑ₙ=1[∞]1/π n; since π is a constant and n increases to infinity, each term in the series gets smaller, but they never reach zero. This is a p-series where p=1, which is known to diverge because the sum approaches infinity as more terms are added.

Series B is given by ∑ₙ=0[∞]1/(e+π)² n, which takes the form of a geometric series with a common ratio less than 1 (specifically 1/(e+π)², which is less than 1). Because the common ratio is less than 1, this geometric series converges.

Therefore, Series B converges, while Series A diverges.

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