For sum 1/n^3 from n=1 to infinity, which test gives a decisive result directly?

• The p-series rule, since p = 3
• The ratio test, since its limit is below 1
• The root test, since its limit is below 1

For a positive-term series, a ratio-test limit of 1 proves convergence.

The series with terms 1/[n sqrt(n)] is a p-series. Enter p as a decimal.

For sum 1/sqrt(n) from n=1 to infinity, which conclusion follows from the p-series rule?

• It converges because its denominator grows without bound.
• It converges because its terms approach zero as n grows.
• It diverges because its exponent p is 1/2 <= 1.
• It diverges because its terms approach a positive constant.

For sum n!/n^n from n=1 to infinity, which test and calculated limit prove convergence?

• The p-series rule, treating 1/e as its fixed exponent.
• The integral test, treating 1/e as the integral's value.
• The ratio test, with consecutive ratio limit 1/e.

For a_n=[(2n+1)/(3n+4)]^n, n >= 1, what does the root test show about sum a_n?

• The root limit is 2/3, so the series converges.
• The root limit is 3/2, so the series diverges.
• The root limit is 1, so the test is inconclusive.
• The root limit is 2/3, so the test is inconclusive.

For n >= 3, 1/(n^2-4) > 1/n^2. Nia says this bound proves convergence because sum 1/n^2 converges. What does this bound establish?

• It proves convergence because the benchmark is convergent.
• It proves divergence because the terms exceed the benchmark.
• It proves convergence because both denominators are positive.
• It gives no convergence conclusion from this lower bound.

Why does sum 1/(n^2+5), from n=1 to infinity, converge?

• Ratio test: its consecutive ratio tends to 1/5 < 1.
• Direct comparison: 0 < 1/(n^2+5) < 1/n^2.
• Term test: its terms tend to zero, proving convergence.

For a_n=1/n, Ivo writes a_(n+1)/a_n=1/(n+1), gets limit 0, and concludes convergence. Which correction is valid?

• The ratio is n/(n+1); its limit 1 is inconclusive.
• The ratio is 1/[n(n+1)]; its limit 0 proves convergence.
• The ratio is (n+1)/n; its limit 1 proves divergence.

For sum n^2/2^n from n=1 to infinity, which test and result are valid?

• Term test: the terms tend to zero, so it diverges.
• Ratio test: its limit is 1/2, so it converges.
• p-Series rule: its exponent is 2, so it converges.