hi
i need help !
a) prove that if ∑ a_n (n=1 to infinity) converges absolutely, then ∑ (a_n)^2 (n=1 to infinity) converges.
b) if ∑ a_n (n=1 to infinity) converges, does ∑ (a_n)^2 (n=1 to infinity) necessarily converge?
i need help !
a) prove that if ∑ a_n (n=1 to infinity) converges absolutely, then ∑ (a_n)^2 (n=1 to infinity) converges.
b) if ∑ a_n (n=1 to infinity) converges, does ∑ (a_n)^2 (n=1 to infinity) necessarily converge?
-
a) For n sufficiently large, we have |a_n| < 1.
So, (a_n)^2 < |a_n| for sufficiently large n.
Since ∑ |a_n| converges (as ∑ a_n converges absolutely), we conclude that ∑ (a_n)^2 also converges by the Comparison Test.
b) This is false; consider ∑ (-1)^n / n^(1/4), which only converges conditionally, but
∑ [(-1)^n / n^(1/4)]^2 = ∑ 1/n^(1/2) is a divergent p-series.
I hope this helps!
So, (a_n)^2 < |a_n| for sufficiently large n.
Since ∑ |a_n| converges (as ∑ a_n converges absolutely), we conclude that ∑ (a_n)^2 also converges by the Comparison Test.
b) This is false; consider ∑ (-1)^n / n^(1/4), which only converges conditionally, but
∑ [(-1)^n / n^(1/4)]^2 = ∑ 1/n^(1/2) is a divergent p-series.
I hope this helps!
-
X = 7