<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] You can’t do this step
$$\lim_{n\rightarrow \infty } \frac{ \frac{1^4}{n^4} + \frac{2^4}{n^4} +...+1 }{n}\color{red}{=\frac{0 + 0 +0 +...+1 }{n}= 0} $$
since you have infinitely many terms which tend to 0 and their sum not necessarily is $0$.
Think to $\sum_1^{\infty} \frac1k$ which diverges.
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