Using the definition

Task number: 2929

Using the definition of the sum of a series, solve the following exercises.

  • Variant 1

    Show that the harmonic series \(\displaystyle \sum_{n=1}^{\infty}\frac1n \) diverges.

  • Variant 2

    Show that the series \(\displaystyle \sum_{n=1}^{\infty} \frac1{\sqrt n}\) diverges.

  • Variant 3

    Investigate the convergence or divergence of the series \(\displaystyle \sum_{n=1}^{\infty} \ln\left(1+\frac1n\right)\).

  • Variant 4

    Let \(\displaystyle \lim_{n\to \infty} a_n = a \in \mathbb R\). Determine \(\displaystyle \sum_{n=1}^{\infty} (a_{n+1}-a_n) \).

Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Cs translation
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