Question

Prove the ratio test for absolute convergence. That is, let $\sum _{k=1}^{\mathrm{\infty }} b_k$be a series with nonzero terms, and let localid="1650855913677" $\rho =\underset{k\to \mathrm{\infty }}{lim} \left|\frac{b_{k+1}}{b_k}\right|$

(a) Show that if ρ < 1, the series converges absolutely.

(b) Show that if ρ > 1, the series diverges.

(c) Show that the test fails when ρ = 1, by finding a convergent series $\sum _{k=1}^{\mathrm{\infty }} c_k$such $\left|\frac{c_{k+1}}{c_k}\right|\to 1$as $k\to \infty$

Changing the order of the summands in a conditionally convergent series can change the value of the sum. We showed this earlier in the section for the alternating harmonic series

Short answer

a) It is proved that if ρ < 1, the series converges absolutely.

b) It is proved that if ρ > 1, the series diverges.

c) proved

Step-by-step solution

Step 1. Given information

$\rho =\underset{k\to \mathrm{\infty }}{lim} \left|\frac{b_{k+1}}{b_k}\right|$

Step 2. Solve for (a)

Therefore,

Now,

The below series is convergent,

Therefore,

Step 3. Solve for (b)

Step 4. Solve for (c)

Now, if,

the test fails as the series can be divergent and can be convergent.