Lecture 17

Question 1

What is the abscissa of absolute convergence

Answer 1

Suppose that the series ∑∣f(n)/(n^s) ∣ does not converge for all s or diverge for all s. Then there exists a real number σ_a, called the abscissa of absolute convergence, such that the series ∑ f(n)/(n^s) converges absolutely if σ > σ_a but does not converge absolutely if σ < σ_a

Question 2

If ∑∣f(n)/(n^s)∣ converges everywhere how do we define σ_a

Answer 2

σ_a = −∞

Question 3

If ∑∣f(n)/(n^s)∣ converges nowhere how do we define σ_a

Answer 3

σ_a = +∞.

Question 4

If N ≥ 1 and σ ≥ c > σa what is the upper bound of
* ∣∑^∞(n=N) f(n)/(n^(−s))∣

Answer 4

N^(−(σ−c)) ∑^∞(n=N) ∣f(n)∣/n^c