Unlock the Power of Convergence: Master the Ratio Test and Geometric Series for Math Mastery

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The proof is left as an exercise. Note that although the comparison test is very useful, it requires knowledge of some convergent series \(\sum_{n=0}^{\infty}a_{n}\) ahead of time. One of the most useful series for comparison is the geometric series \(\sum_{n=0}^{\infty}ar^{n}\). In the case \(|r|<1\), this series converges and the sum is \(\sum_{k=0}^{\infty}ar^{n}=\dfrac{a}{1-r}\).

**Theorem 24** (Ratio Test).: _Suppose \(\sum_{n=0}^{\infty}a_{n}\) is a series of nonzero complex numbers and suppose the limit_

\[r=\lim_{n\to\infty}\left|\dfrac{a_{n+1}}{a_{n}}\right|\]

_exists, then_

* _If_ \(r<1\)_,_ \(\sum_{n=0}^{\infty}a_{n}\) _converges absolutely.__._ 2. _If_ \(r>1\)_,_ \(\sum_{n=0}^{\infty}a_{n}\) _diverges._
3. _If_ \(r=1\)_, the test is inconclusive._

Proof

: The result is clear when \(r>1\). For the case when \(r=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_{n}}\right|<1\), if \(\lambda\) satisfies \(r<\lambda<1\), then there exists \(N\in\mathbb{N}\) such that

\[\frac{|a_{n+1}|}{|a_{n}|}<\lambda\quad\text{for all}\quad n\geq N.\]

Therefore,

\[|a_{n}|\leq|a_{N}|\lambda^{n-N}\quad\text{for all}\quad n\geq N.\]

Now apply the comparison test. The test gives no information when \(r=1\). For example, the series \(\sum_{n=0}^{\infty}\frac{1}{n}\) and \(\sum_{n=0}^{\infty}\frac{1}{n^{2}}\) both have \(r=1\), while the first one (harmonic series) diverges, but the second one converges.

**Theorem 25** (Root Test).: _Suppose \(\sum_{n=0}^{\infty}a_{n}\) is a series of complex numbers, and set_

\[r=\lim_{n\to\infty}\sqrt[n]{|a_{n}|}.\]

_Then_

1. _If_ \(r<1\)_,_ \(\sum_{n=0}^{\infty}a_{n}\) _converges absolutely._ 2. _If_ \(r>1\)_,_ \(\sum_{n=0}^{\infty}a_{n}\) _diverges._
3. _If_ \(r=1\)_, the test is inconclusive._

Proof

: If \(r<1\), we can choose \(\lambda\) with \(r<\lambda<1\) and an integer \(N\) so that

\[\sqrt[n]{|a_{n}|}<\lambda\quad\text{for all}\quad n\geq N.\]

Equivalently

\[|a_{n}|<\lambda^{n}\quad\text{for all}\quad n\geq N.\]

Since \(0<\lambda<1\), \(\sum_{n=0}^{\infty}\lambda^{n}\) converges, and the convergence of the series follows from the comparison test. When \(r>1\), choose any number \(\lambda\) with \(1\lambda\) for infinitely many indices \(n\) by the definition of “\(\lim\) sup.” Therefore, \(|a_{n}|>\lambda^{n}\to\infty\) as \(n\to\infty\) through some subsequence. In particular, the general term of the series \(\sum_{n=0}^{\infty}a_{n}\) does not tend to zero; therefore the series is divergent.

**Remark 15**.: The root test is more powerful and has wider scope than the ratio test. However, the ratio test is easier to apply, for example, for series involving factorials. Both tests conclude divergence from the fact that \(a_{n}\) does not tend to zero as \(n\to\infty\).

### Exercises

1. Suppose \(\{x_{n}\}\) and \(\{y_{n}\}\) are sequences such that \(x_{n}\to L\) and \(y_{n}\to M\). Prove that:

1. \((x_{n}+y_{n})\to L+M\).
2. \(cx_{n}\to cL\) for any constant \(c\).

2. Consider the following sequences and using the definition of limit to show that:

1. \(\lim_{n\to\infty}\frac{1}{n}=0\).
2. \(\lim_{n\to\infty}\frac{n}{n+1}=1\).
3. \(\lim_{n\to\infty}\frac{n}{2^{n}}=0\).

3. Let \(f:C\to\mathbb{R}\) be a map, where C is a countable set (recall that C is countable if and only if there exists a bijective mapping \(g:\mathbb{N}\to C\)). Prove that we can express \(f(C)=a_{k}\), where \(a_{k}\in\mathbb{R}\) and \(k\in\mathbb{N}\).

4. Show that if \(\{x_{n}\}\) converges to \(L\), then \(\{|x_{n}|\}\) converges to \(|L|\). What about the converse?

5. Let \(S\) be a nonempty subset of \(\mathbb{R}\) which is bounded above. Let \(s=\sup S\). Show that there exists a sequence \(\{x_{n}\}\) in \(S\) which converges to \(s\).

6. Show that \(\{x_{n}\}\) defined by

\[x_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n}\]

is divergent.

7. Show that \(\{x_{n}\}\) defined by

\[x_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n}-\ln(n)\]is convergent.

8. If \(\{x_{n}\}\) and \(\{y_{n}\}\) are Cauchy sequences in \(\mathbb{R}\), then show that \(\{x_{n}+y_{n}\}\) and \(\{x_{n}y_{n}\}\) are Cauchy sequences in \(\mathbb{R}\).

9. Show that the sequence \(\{x_{n}\}\) defined by

\[x_{n}=\int_{1}^{n}\frac{\cos(t)}{t^{2}}dt\]

is Cauchy.

10. Let \(\{x_{n}\}\) be a sequence such that there exist \(A>0\) and \(C\in(0,1)\) for which

\[|x_{n+1}-x_{n}|\leq AC^{n}\]

for any \(n\geq 1\). Show that \(\{x_{n}\}\) is Cauchy. Is this conclusion still valid if we assume only

\[\lim_{n\to\infty}|x_{n+1}-x_{n}|=0\?\]

11. Show that if a subsequence \(\{x_{n_{k}}\}\) of a Cauchy sequence \(\{x_{n}\}\) is convergent, then \(\{x_{n}\}\) is convergent.

12. Let \(\{x_{n}\}\) be defined by

\[x_{1}=1\ \ \mbox{and}\ \ x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{2}{x_{n}}\right)\.\]

Show that \(\{x_{n}\}\) is convergent and find its limit.

13. Find the limit superior and limit inferior for the following sequences \(\{x_{n}\}\):

1. \(x_{n}=2^{n}\).
2. \(x_{n}=n\).
3. \(x_{n}=1+(-1)^{n}+1/2^{n}\).

14. If \(\{x_{n}\}\) and \(\{y_{n}\}\) are bounded real sequences, show that

\[\limsup_{n\to\infty}(x_{n}+y_{n})\leq\limsup_{n\to\infty}x_{n}+\limsup_{n\to \infty}y_{n}.\]

15. Suppose that \(\sum_{n=0}^{\infty}x_{n}\) is a series of positive terms which is convergent. Show that \(\sum_{n=0}^{\infty}\frac{1}{x_{n}}\) is divergent. What about the converse?