Question

\[\text{ If } a_0+a_1+a_2+\cdots+a_k=0 \\ \text{ show that } \\ \lim_{n \rightarrow \infty } (a_0 \sqrt{n} + a_1 \sqrt{n+1} +a_2 \sqrt{n+2} + \cdots + a_k\sqrt{n+k})=0\]

Answer 1

hmm if i answer this ima be bummed out

Answer 2

\[\text{ So if we assume } k=1 \text{ we have } \\ a_0+a_1=0 \\ a_0=-a_1 \\ \lim_{n \rightarrow \infty} (a_0 \sqrt{n}-a_0 \sqrt{n+1}) \\ a_0 \lim_{n \rightarrow \infty} \frac{n - (n+1)}{\sqrt{n}+\sqrt{n+1}}=a_0 \lim_{n \rightarrow \infty} \frac{\frac{-1}{\sqrt{n}}}{1+\sqrt{1+\frac{1}{n}}}=a_0(0)=0\]

Answer 3

OH GOD THIS IS HARD

Answer 4

but i can still try :)

Answer 5

\[ \text{ So if we assume k=2 we have } \\ a_0+a_1+a_2=0 \\ a_0=-(a_1+a_2) \\ \lim_{n \rightarrow \infty} (a_0 \sqrt{n}+a_1 \sqrt{n+1} +a_2 \sqrt{n+2}) \\ = \lim_{n \rightarrow \infty}(a_1 \sqrt{n+1}-a_1 \sqrt{n}+a_2 \sqrt{n+2}-a_2 \sqrt{n}) \\ = -a_1\lim_{n \rightarrow \infty} (\sqrt{n}-\sqrt{n+1})-a_2 \lim_{n \rightarrow \infty}(\sqrt{n}-\sqrt{n+2})=-a_1(0)-a_2(0)=0-0=0\]

Answer 6

\[\text{ So if we have the general case } \\ a_0=-(a_1+a_2+ \cdots + a_k) \\ \[\lim_{n \rightarrow \infty}(a_1 \sqrt{n+1}-a_1 \sqrt{n}+a_2 \sqrt{n+2}-a_2 \sqrt{n} + \cdots + a_k \sqrt{n+k}-a_k \sqrt{n}) \\ 0+0+\cdots +0=0\]

Answer 7

actually this one wasn't that bad

Answer 8

we can demonstrate your statement using the induction principle as below:
for k=1, your statement is true as you shoved above, also for k=2, your statement is true as you showed above again, so let's suppose that your statement is true for a generic k, and evaluate it for k+1, namely:
for k+1, we have:
\[a _{0}+a _{1}+...+a _{k}+a _{k+1}=0\]
now we can write this:
\[a _{0}\sqrt{n}+a _{1}\sqrt{n+1}+...+a _{k}\sqrt{n+k}+a _{k+1}\sqrt{n+k+1}=\]
\[=a _{0}\sqrt{n}+a _{1}\sqrt{n+1}+...+a _{k}\sqrt{n+k}-a _{0}\sqrt{n+k+1}...\]
\[...-a _{1}\sqrt{n+k+1}-...a _{k}\sqrt{n+k+1}=\]
\[=a _{0}(\sqrt{n}-\sqrt{n+k+1})+...+a _{k}(\sqrt{n+k}-\sqrt{n+k+1})=\]
\[a _{0}\frac{ -k-1 }{ \sqrt{n} +\sqrt{n+k+1}}+...a _{k}\frac{ -1 }{ \sqrt{n+k}+\sqrt{n+k+1} }\]
and the last quantity to goes to zero as n goes to infinity