Understanding Direct Comparison Tests

The video explains the direct comparison test for series convergence and divergence. It covers the basic principles, including when the test is conclusive and when it is not. Several examples illustrate how to apply the test, highlighting cases of convergence, divergence, and inconclusive results. The video emphasizes understanding the test's limitations and suggests using the limit comparison test when necessary.

What is the basic idea behind the direct comparison test?

• If a smaller series converges, the larger series must diverge.

• If a larger series diverges, the smaller series must converge.

• If a larger series converges, the smaller series must converge.

• If a smaller series diverges, the larger series must converge.

Which statement is true about the reverse of the direct comparison test?

• If the small series converges, the big series must converge.

• The reverse statements are inconclusive.

• The reverse statements are always true.

• If the big series diverges, the small series must diverge.

In the example of the series 1/(4 + 3^n), which term is considered insignificant when n is large?

• n

• 3^n

• 4

• 1

What type of series is 1/3^n?

• Arithmetic series

• Geometric series

• Harmonic series

• P-series

For the series 1/(n^3 + 5), what is the type of the simplified series?

• Arithmetic series

• Harmonic series

• P-series

• Geometric series

In the example of 1/(4 + cube root of n), what is the value of p in the p-series?

• 2

• 1/3

• 3

• 1

What is the common ratio of the geometric series 2^n/7^n?

• 1/7

• 7/2

• 1/2

• 2/7

In the series ln(n)/n, what is the significance of n compared to ln(n) as n approaches infinity?

• n increases faster than ln(n)

• ln(n) is always greater than n

• Both increase at the same rate

• ln(n) increases faster than n

What is the p-value for the series 1/n?

• 1

• 0

• 3

• 2

In the example 1/sqrt(n^4 + 5), what is the simplified form of the denominator?

• sqrt(n^4)

• n^4

• n

• n^2