The sequence $\phi_n= 1+\frac{1}{2^4}+\frac{1}{3^4}+\ldots+\frac{1}{n^4}$ is bounded above . A sequence $(\phi_n)$ is defined as follows : $$\phi_n= 1+\frac{1}{2^4}+\frac{1}{3^4}+\ldots+\frac{1}{n^4}$$
Show that the sequence is convergent.
Because this sequence is monotonic, proving it is bounded above will be sufficient to prove that it is convergent.
So how to show that this sequence is bounded above?

Since $t \mapsto \dfrac{1}{t^4}$ is decreasing, we have
$$\frac{1}{(k+1)^4}=\int_{k}^{k+1}\frac{dt}{(k+1)^4}\leq \int_{k}^{k+1}\frac{dt}{t^4},\,\,\, k\geq1,$$ which gives$$\phi_n= 1+\frac{1}{2^4}+\frac{1}{3^4}+\ldots+\frac{1}{n^4}=1+\sum_{k=1}^{n-1}\frac{1}{(k+1)^4}\leq 1+\int_{1}^{n}\frac{dt}{t^4}=\frac{4}{3}-\frac{1}{3n^3}<\frac{4}{3}.$$
