About little-o notation

Question

What is $x_{n}$ in the following equation: $n^4$(1+o(1))+$x_{n}$=$n^4$(1+o(1))

It is o($n^4$), but I don't understand, why? Can you please explain this to me?

Answer 1

It's quite simple. Expand both sides:

• $n^4(1+o(1))=n^4+n^4o(1)=n^4+o(n^4)$
• so the equality can be rewritten as $$n^4+o(n^4)+x_n=n^4+o(n^4)\iff x_n=o(n^4)-o(n^4)=o(n^4).$$