does the series the sum from n=1 to infinity of 4/((squqre root n)+2)

Question

4/(√x +2)

Answer 1

I assume you want to see if the summation Σ[n=1,∞] 4/(√n+2) converges.

Since Σ[n=1,∞] 4/(√n)dx = Σ[n=1,∞] 4/n^1/2, and 1/2<1, then Σ[n=1,∞] (4/√n) diverges by the p-test

Let a_n=4/(√n+2) and b_n=4/(√n). By the Direct Comparison test, 0≤a_n≤b_n must hold for either of the following conditions:

If Σb_n converges, then Σa_n converges OR If ∑a_n diverges, then ∑b_n diverges

The problem however is that because Σb_n diverges, which is the larger series, we can't use the Direct Comparison Test to tell if the smaller series Σa_n diverges. We can try using the limit comparison test:

Define c = lim_n->∞ a_n/b_n. If c>0 and c<∞, then both series will either converge or diverge.

In this case, c = lim_n->∞ [4/(√n+2)]/[4/(√n)] = 1, so both series will diverge since we already stated that ∑b_n=∑[n=1,∞] 4/(√n) diverges. Therefore, Σ[n=1,∞] 4/(√n+2) diverges.

Hope this helped!