Question

15. Soru 4 Pu Which of the following gives the complete list of the convergent series among I, II, and III? I. II. III. 8W 1 1 1 n=1 n n=1 nn ns1 n3 All and 111 O B) III only C) I and II D) Il only E) I and III

Answer 1

In=1 n Cauchy's Convergence Condition: Xan converges, if, and only if For every e > 0 there is a natural number N such that|an+1 <E, V n> N and p > 1 + An+2 + + An+p 1 1 1 12=15-ah + + + Al n + 2 2n Taking S2n -Sn = n n+1 Therefore there cannot be found a number N that satisfies Cauchy's condition 2n - = diverges

n=1 1 2 n Apply p - Series Test: ΣΗ- n p - Series Test: If the series is of the form Lon=11, where p > 0 n If p > 1, then the p - series converges If o<p < 1, then the p-series diverges p = 2, p > 1, by the p - Series test criteria = converges

2 and 3

n=1 al- n - Apply p - Series Test: En=1 n P-Series Test: If the series is of the form =1 where p > 0 n If p > 1, then the p - series converges If 0 <p < 1, then the p - series diverges p = 3, p > 1, by the p - Series test criteria = converges

Option A