SUM(arctan(a_n) converges if and only if SUM(a_n) converges.
a_n in this case is a sequence of positive numbers.
[quote=Juancd08;337083]SUM(arctan(a_n) converges if and only if SUM(a_n) converges.
preposition
1.Let arctan(a_n)=b_n => tan(b_n)=a_n
2.ıf any sequence is converges than its subsequences also converge.
Solution
tan(b_n) is a subsequence of b_n. By the comparison princeple if b_n is converge then tan b_n also converge. Conversly now show that if tan(b_n) is converge than b_n converges. Similiarly arctan(a_n) is subsequence of a_n then right handed also satisfies. then proof is done
[QUOTE=mathemanyak;338106]How is tan(b_n) a subsequence of b_n? You are saying that, for any positive integer m, there exist a positive integer n such that b_n= tan(b_m)? I don't see that. Or are you simply asserting that tan(b_n) is dominated by bn: that tan(b_n)< b_n for all n?