1. Evaluate 2+4/7+8/49+16/343+... .
2. Evaluate 1+i/2-1/4-i/8+1/16+i/32-...(where i is the imaginary unit). Express your answer in the form of a+bi, where a and b are real.
+129840 1. Evaluate 2+4/7+8/49+16/343+... .
We have
2 + the sum of an infinite geometric series where 4/7 is the first term and the common ratio = 2/7 .......so.....
2 + (4/7) / [1 - 2/7] = 2 + (4/7) / (5/7) = 2 + 4/5 = 14/5 = 2.8
2. Evaluate 1+i/2-1/4-i/8+1/16+i/32-...(where i is the imaginary unit). Express your answer in the form of a+bi, where a and b are real.
We can write this as
( 1 - 1/4 + 1/16 - 1/64 + .......) + ( i/2 - i/8 + i/32 - i/128 +.........) =
∞ ∞
∑ (- 1/4) n + i ∑ (1/2)2n + 1 (-1)n =
n = 0 n = 0
4/5 + 2/5 i
