Notes PDF
Introductory Problems
Determine if the series a1 + a2 + a3 + a4 + …converges or diverges, where:
1. an = sin(π/n)
Answer
1. The series diverges.
Solution
1. Recalling sin(x)/x → 1 as x → 0, we see sin(π/n)/(π/n) → 1 as n → ∞ Because ∑π/n is just π times the harmonic series, which diverges, the given series diverges by the Limit Comparison Test.
2. an= 1/(1 + 2 + … + n)
Answer
1. The series converges.
Solution
1. Recall 1 + 2 + … + n = n(n+1)/2. (Using mathematical induction, this can be proved easliy.) Note that 2/(n2 + n) < 2/n2, which is twice a convergent p-series, so the original series converges by the Comparison Test.
3. an = (1 + cos(n))/n3
Answer
1. The series converges.
Solution
1. Because -1 ≤ cos(n) ≤ 1, we see 0 ≤ 1 + cos(n) ≤ 2. Consequently, 0 ≤ (1 + cos(n))/n3 ≤ 2/n3. The original series converges by comparison to a convergent p-series.