Question

For each of the following, identify whether the sum converges to a real number or if it diverges. If it converges, find a closed form for the real number it converges to. You may cite results stated in class or in homework. a.) 1 +(12) + ((?) + ((147)* + (1/21/2((??))" + 2! 3! b.) 1+(143) + (142) + (13?) + (143) + ...

Answer 1

a.) Applying the geometric series formula, we have:

1 + (1/2) + (1/4) + (1/8) + ... = 2

Since the ratio is 1/2 and the first term is 1, the sum converges to 2.

b.) We can rewrite this series as:

1 + 3/2 + 4/3 + 3/4 + 4/5 + ...

The terms of this series do not approach 0 and the partial sums do not appear to be bounded. Therefore, by the Divergence Test, the series diverges to infinity.