Calculus 2 question

Question

Is the series convergent or divergent

series goes form 0 to infinite

(3^n-1 + pi^n+1)/2^2n

Answer 1

The sequence converges because the terms are all positive and all less than 4.25.

I am not sure about the sum (series).

If I get the answer later, I will get back to you.

LATER

OK, the sequence converges but not for the reason I gave at first.

a_n+1 - a_n < 0, i.e. the sequence is decreasing and it is bounded below by zero; therefore, the sequence converges to 0. (Do the subtraction!)

lim (as n -> ∞) a_n+1 / a_n =π/4 < 1; therefore the series (sum) converges by the ratio test.

(Set up the ratio; divide the powers of 2 so that you have a factor of 1/4 in the ratio.

Then divide the numerator and denominator by πⁿ⁺¹ and you will see the limit.)

Please note:I set up a spreadsheet for the first 51 terms. It appears that the sum converges to 16, but I cannot prove that.