Divergence of Harmonic Series and Sequence Behavior at Infinity

Let $(H_n)$ the sequence defined for $n\in \mathbb{N}$ by $H_n= \sum_{k=1}^n \frac 1k.$

Prove that $H_n\xrightarrow[n\to+\infty]{}+\infty$ .
Let $(u_n)$ a sequence such that $n(u_{n+1}-u_n)\xrightarrow[n\to+\infty]{}1$ . Prove that $u_n\xrightarrow[n\to+\infty]{}+\infty$ .

1. Knowing

\ln(1+x)\le x

, we have

\frac1k\ge \ln\left(1+\frac1k\right)=\ln(k+1)-\ln k

H_n\ge \sum_{k=1}^n \ln(k+1)-\ln k= \ln(n+1)

and then

H_n\xrightarrow[n\to+\infty]{}+\infty

.

2. There exists

N\in \mathbb{N}

such that for all

n\ge N

n(u_{n+1}-u_{n})\ge 1/2.

Hence

u_{n+1}-u_{N}\ge \sum_{k=N}^n u_{k+1}-u_{k}\ge \frac12 \sum_{k=N}^n\frac1k= \frac12(H_{n}-H_{N-1})\xrightarrow[n\to+\infty]{}+\infty

and then

u_{n}\xrightarrow[n\to+\infty]{}+\infty

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