Using only the divergence test, determine whether or not the following series diverges
Applying the divergence test yields
Since the limit equals zero, the divergence test yields no conclusion.
Explanation of Each Step
To apply the divergence test, we replace our sigma with a limit.
To apply our limit, a little algebraic manipulation will help: we may divide both numerator and denominator by the highest power of k that we have. Taking the radical into account, the highest power of k is 1, so we divide both numerator and denominator by k1 = k.
The algebra in the denominator may be a little tricky. Here is the above derivation with two extra lines of math:
Potential Challenge Areas
Drawing the Wrong Conclusion
As mentioned earlier, one mistake students could make either one of two mistakes when the limit equals zero:
- that the series converges
- that the convergence of the given series cannot be established
However, when the limit equals zero, the test yields no conclusion, and it could be that the convergence of the given series could be established with a different test. Be careful to not make either of these mistakes.