Urgent Question: Convergens of series
Could somebody here please verify my proof?
Given the two sequences $\displaystyle \sum_{n=1}^{\infty} c_{n}$ and $\displaystyle \sum_{n=1}^{\infty} d_{n}$ which is defined as
$\displaystyle d_{n} = \frac{c_1 + c_2 + \ldots + c_n}{n}$
1) show that if $\displaystyle c_{n} \rightarrow 0$, for $\displaystyle n \rightarrow \infty$ is true. Then $\displaystyle d_{n} \rightarrow 0$ for $\displaystyle n \rightarrow \infty$
2) show that if $\displaystyle c_{n} \rightarrow c$, for $\displaystyle n \rightarrow \infty$. Then $\displaystyle d_{n} \rightarrow c$ for $\displaystyle n \rightarrow \infty$
Solution 1)
I choose an $\displaystyle N' \in \mathbb{N}$ which for all m, $\displaystyle m \geq N'$ implies that $\displaystyle |c_m|< \frac{\epsilon}{2}$.
I then choose a random $\displaystyle n > N'$ there is an N'' such that for all
from which is follows that
$\displaystyle d_{n} = \frac{c_1 + c_2 + \ldots + c_N'}{n} + \frac{c_{N'+1}+ c_2 + \ldots + c_n}{n}$
Then by the triangle inequality
$\displaystyle \begin{array}{cccc}|d_n| &\leq& |\frac{c_1 + c_2 + \ldots + c_N'}{n}| + |\frac{c_{N'+1}+ c_2 + \ldots + c_n}{n}| \\
|d_n| &\leq& \frac{|c_1 + c_2 + \ldots + c_N'|}{n} + \frac{|c_{N'+1}|+ |c_2| + \ldots + |c_n|}{n} \end{array} \\
|d_n| &\leq& \frac{|c_1 + c_2 + \ldots + c_N'|}{n} + \frac{(N-N')\frac{\epsilon}{2}}{2} \\ |d_n| &\leq& \frac{|c_1 + c_2 + \ldots + c_N'|}{n} + \epsilon \end{array}\end{array}
$
there is an N'' such that for all $\displaystyle n \geq N''$, thus $\displaystyle |d_n| < \epsilon$
Solution 2)
$\displaystyle y_n:=c_n-c \rightarrow 0 \mathrm{ as }n \rightarrow \infty \mathrm{ and thus } d_n-c=\frac{(c_1-c)+\cdots +(c_n-c)}{n}=\frac{y_1+\cdots +y_n}{n} $
will tend to 0 by part (a).
I have let the two last tex modules stand because I cannot get them to work with "math", "/math"
Best regards
