Question

Evaluate using root test.

Infinity

Summation sign (1- 1/k)^k

k = 1

Answer 1

$lim _{k\to \infty}((1-1/(k+1))^{k+1}/(1-1/k)^k)$

$lim _{k\to \infty}((k/(k+1))^{k+1}/((k-1)/k)^k)$

$lim _{k\to \infty}((k^2/(k^2-1))^{k})$

$lim _{k\to \infty}((1/(1-1/k^2))^{k})$ is of the form $1^{\infty}$

$e^{lim _{k\to \infty}(((k^2/(k^2-1))-1)k)}$

$e^{lim_{k \to \infty}(k/(k^2-1))}$

$e^0=1$

Root test fails as limit value is equal to 1.