## More Examples on Series

**Problem 1:** Test for convergence

.

**Answer:** Since we have a power n in the series, we will use the
Root-Test. Set

.

We have

.

Since

,

and

,

we get

.

But,

.

Hence,

.

Therefore, by the Root-Test, the series

is divergent.

**Problem 2:** Test for convergence

.

**Answer:** The sum of two series converges, if both of the sums converge.
Hence the series

will diverge, if we can show
that
diverges, while the series
converges.
Since
=

,

and
and the series
diverges by the *p*-test, we
conclude that
diverges.
On the other hand,

converges by
the ratio test:
This establishes that
diverges.
**Problem 3:** Test for convergence

.

**Answer:** We will use the Ratio-Test (try to use the Root-Test to see how difficult it is). Set

.

We have

.

Algebraic manipulations give

,

since

.

Hence, we have

,

which implies

.

Since , we conclude, from the Ratio-Test, that the series

is convergent.

**Problem 4:** Determine whether the series

is convergent or divergent.

**Answer:** Consider the function

.

It is easy to
check that *f*(*x*) is decreasing on . Hence, for any , we have for any ,

,

which implies

,

that is,

.

Using this inequality, we get

,

since

.

Since

,

we deduce that the partial sums associated to the series

are not bounded. Therefore, the series

is divergent.

**Remark:** Note that the proof given above is the proof of the Integral-Test. In other words, we may have just used to Integral-Test to get the conclusion. Also, the series given here is part of a type of series called Bertrand series defined as

.

**
**

**
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