<s>[INST] limit without expansion I was solving this limit  instead of using sum of expansion i  did this and got zero  but answer is 1/5 by expansion why this is wrong
$$\lim_{n\rightarrow \infty } \frac{1^4 + 2^4 + \ldots +  n^4}{n^5}$$
$$\lim_{n\rightarrow \infty } \frac{n^4( \frac{1^4}{n^4} + \frac{2^4}{n^4} +\ldots + 1 )}{n^5}$$
$$= \lim_{n\rightarrow \infty } \frac{( \frac{1^4}{n^4} + \frac{2^4}{n^4} +\ldots + 1 )}{n}$$
$$= \frac{(0 + 0 +0 \ldots + 1 )}{n} = 0$$ this is how i got  ,
 [/INST] Also, you can use Stolz: https://en.wikipedia.org/wiki/Stolz%E2%80%93Ces%C3%A0ro_theorem
$$\lim_{n\rightarrow+\infty}\frac{1^4+2^4+...+n^4}{n^5}=\lim_{n\rightarrow+\infty}\frac{n^4}{n^5-(n-1)^5}=\frac{1}{5}.$$
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