Question

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

∑ₙ=1[infinity](e⁷ / n-e⁷ /(n+1))

Answer 1

The series ∑ₙ=1[∞](e⁷ / n-e⁷ /(n+1)) is a telescoping series that converges, with the nth partial sum approaching e⁷ as n approaches infinity.

Step-by-step explanation:

Determine Convergence or Divergence of a Series

To determine whether the series ∑ₙ=1[∞](e⁷ / n-e⁷ /(n+1)) is convergent or divergent, we can express the nth partial sum, sₙ, as a telescoping sum. A telescoping series is one where terms cancel out except for the first few and last few terms. For this particular series, we can write:

sₙ = (e⁷ / 1 - e⁷ / 2) + (e⁷ / 2 - e⁷ / 3) + ... + (e⁷ / n - e⁷ / (n+1))

Most terms cancel out, leaving us with:

sₙ = e⁷ - e⁷ / (n+1)

As n approaches infinity, e⁷ / (n+1) approaches 0, which means the series converges to e⁷.