Root test & Ratio test

**adam63** · March 12th 2010, 11:12 AM

Ratio test : first of all, if $\frac{a_{k+1}}{a_k} \rightarrow \infty$ , then the sequence does not converge, which means the series $\sum a_k$ surely won't converge.

Same with the root test - if limsup $(a_k)^\frac{1}{k} \rightarrow \infty$ , then $a_k \rightarrow \infty$ , and the sequence diverges, so the series will diverge too.

**kingwinner** · March 13th 2010, 03:52 PM

Originally Posted by adam63

Ratio test : first of all, if $\frac{a_{k+1}}{a_k} \rightarrow \infty$ , then the sequence does not converge, which means the series $\sum a_k$ surely won't converge.

Same with the root test - if limsup $(a_k)^\frac{1}{k} \rightarrow \infty$ , then $a_k \rightarrow \infty$ , and the sequence diverges, so the series will diverge too.

So it looks like the root test and ratio test is applicable for the case c=∞ and r=∞ as well. (and we can say c=∞>1 or r=∞>1, hence the series diverges)

But I don't understand why your claims are true. Can you explain/justify?

Thank you!

**adam63** · March 14th 2010, 03:31 AM

a series can converge ONLY if the sequence converges.
Then, if the sequence diverges, the series will surely diverge

.

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