Ratio Test for Integrals
If and , then converges exactly when converges.
Proposition 4.33 (Ratio Test for Integrals)
Let satisfy , and
then
Limiting cases
The hypothesis cannot be relaxed to or without losing an equivalence:
- If , then for very large , so by the comparison test from large onwards,
- If , then for very large , and similarly
Examples
- For , and , so by the limiting case above,
- converges: since
and , the ratio test gives .
Related
Stated in
- Proposition 4.33 (Ratio Test for Integrals)§4.5 Improper Integrals