converge

• April 24th 2010, 11:22 AM
sfspitfire23
converge
If $a_n$ converges to 0, does $\frac{a_n}{2}$ converge to 0?
• April 24th 2010, 11:31 AM
Krizalid
yes.
• April 24th 2010, 11:34 AM
sfspitfire23
thx
• April 24th 2010, 12:02 PM
harish21
Quote:

Originally Posted by sfspitfire23
is there a theorem that tells me if the series of a_k is convergent then the series of a_k/2 is convergent?

The quotient property of convergent sequences: If a sequence $a_n$ converges to a number a, and the sequence $b_n$ converges to a number b, with $b_n$ not equal to 0 for all indices n, and b is not equal to 0, then the sequence of quotients $\frac{a_n}{b_n}$ converges to the quotient $\frac{a}{b}$.

$lim (n\rightarrow \infty) [\frac{a_n}{b_n}] = lim (n\rightarrow \infty) [a_n \times \frac{1}{b_n}]= a \times \frac{1}{b} = \frac{a}{b}$