Question

Determine whether the series is convergent or divergent by expressing sₙ as a telescoping sum

∑ₙ=2[infinity](e¹0 /(n+1)-e¹0 / n)

Answer 1

The given series can be expressed as a telescoping sum, where each term cancels out with the next term. Therefore, the series converges to a specific value.

Step-by-step explanation:

To determine whether the series is convergent or divergent, we need to express the sum sₙ as a telescoping sum. The given series is ∑ₙ=2[∞] (e¹⁰ /(n+1)-e¹⁰ / n). Let's start by simplifying the terms:

e¹⁰ /(n+1) - e¹⁰ / n = (e¹⁰n - e¹⁰(n+1))/(n(n+1)) = -e¹⁰/(n(n+1))

Now, we can rewrite the series as ∑ₙ=2[∞] (-e¹⁰/(n(n+1))). This is a telescoping sum because each term cancels out with the next term. Let's write out the terms:

-e¹⁰/(2(2+1)) + e¹⁰/(2(2+1)) - e¹⁰/(3(3+1)) + e¹⁰/(3(3+1)) - ...

As we can see, each term cancels out with the next term, leaving us with only the first and last terms of the series. Therefore, the series converges to -e¹⁰/(2(2+1)).