Question

Divergent or convergent? I understand the testing method but is confused by how to use it

Answer 1

\[\int\limits_{-\infty}^{0}2^rdr\]\[\int\limits_{1}^{\infty}\frac{ lnx }{ x^3 }\]

Answer 2

\(\large \mathbb{\int\limits_{-\infty}^{0}2^rdr} = \lim \limits_{t \to -\infty} \int_t^0 2^r dr = \lim \limits_{t \to -\infty} (\frac{2^r}{\ln 2}\Big|_t^0) = \frac{1}{\ln 2}\lim \limits_{t \to -\infty} (1-2^t) \)

Answer 3

take the limit

Answer 4

\(\large \mathbb{ \frac{1}{\ln 2}\lim \limits_{t \to -\infty} (1-2^t) = \frac{1}{\ln 2}(1-2^{-\infty}) = \frac{1}{\ln 2} (1-0) = \frac{1}{\ln 2} }\)

Answer 5

So, it is convergent. it covnerges to \(\frac{1}{\ln 2}\)