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  ,

The problem is that while each term tends to zero, the number of terms tends to infinity at the same time.
The rule “the limit of a sum is the sum of the limits” only applies if you have a finite sum with a fixed number of terms (each of which has a finite limit).
(Also, you can't really write
$$\lim_{n \to \infty} (\dots) = (\text{some expression involving $n$})$$
since there can be no $n$ left in your expression after you have let $n \to \infty$. The symbol $n$ stands for the same number everywhere, so you can't selectively let some $n$ tend to $\infty$ while leaving other $n$ unaffected.)
