converges due to the comparison test.

We start with the equation. Sincefor all values of k, we can multiply both side of the equation by the inequality and getfor all values of k. Sinceis a convergent p-series with,hence also converges by the comparison test.

We will use the Limit Comparison Test to show this result.

We first denote the genera term of the series by:

and.

We have和本系列具有相同的性质。

We know that

is convergent by comparing the integral

which we know is convergent.

Therefore by the Limit Comparison Test.

we have.

If the series converges, then we know the terms must approach zero. At some point, the terms will be less than 1, meaning when you take the third power of the term, it will be less than the original term.

Other answers are not true for a convergent series by theterm test for divergence.

In addition, the limit of thepartial sums refers to the value the series converges to. A convergent series need not converge to zero. The alternating harmonic series is a good counter example to this.

cannot be an interval of convergence because a theorem states that a radius has to be either nonzero and finite, or infinite (which would imply that it has interval of convergence). Thus,can never be an interval of convergence.

The divergence tests states for a series, ifis either nonzero or does not exist, then the series diverges.

The limitdoes not exist, so therefore the series diverges.

To prove the series converges, the following must be true:

Ifconverges, thenconverges.

Now, we simply evaluate the limit:

The shortcut that was used to evaluate the limit as n approaches infinity was that the coefficients of the highest powered term in numerator and denominator were divided.

The limit approaches a number (converges), so the series converges.

First, we reduce the series into a simpler form.

我们知道这个级数收敛,因为

By the Geometric Series Theorem, the sum of this series is given by

Notice how this series can be rewritten as

Therefore this series diverges.

Infinite series can be added and subtracted with each other.

Since the 2 series are convergent, the sum of the convergent infinite series is also convergent.

Note: The starting value, in this case n=1, must be the same before adding infinite series together.

This is a fundamental property of series.

For any constant c, ifis convergent thenis convergent, and ifis divergent,is divergent.

is divergent in the question, and the constant c is 10 in this case, sois also divergent.