Consider the series 1−1+1−1+.... Write the series using sigma notation. and

Question

Write down the first four terms in the sequence of partial sums, and then find the exact formula for the N-th partial sum.

Use the exact formula to determine if the series converges or diverges. If it converges, find the value it converges to.

Answer 1

1-1+1-1 = 0 for the first 4 terms

for any odd number of terms, the partial sum = 1

for any even number of terms the partial sum =0

It's a geometric sequence with common ratio = r = -1

partial sum of n terms = Sn = a1(1-r^n)/(1-r), with a1 = 1

Sn = (1)(1-r^n)/(1-r)

Sn = (1-(-1)^n)/2

S4 = (1-(-1)^4)/2 = (1-1)/2 = 0

the sum never converges It oscillates between 1 and 0. That's the standard answer.

(But as a minor footnote, prolific mathematician Euler would argue the sum of infinite terms =1/2. Bernoilli too.)