
# 4.E: Convergence of Sequences and Series (Exercises)

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### Q1

Prove that if $$\lim_{n \to \infty }s_n = s$$ then $$\lim_{n \to \infty }\left |s_n \right | = \left |s \right |$$. Prove that the converse is true when $$s = 0$$, but it is not necessarily true otherwise.

### Q2

1. Let $$(s_n)$$ and $$(t_n)$$ be sequences with $$s_n ≤ t_n,∀n$$. Suppose $$\lim_{n \to \infty }s_n = s$$ and $$\lim_{n \to \infty }t_n = t$$. Prove $$s ≤ t$$. [Hint: Assume for contradiction, that $$s > t$$ and use the deﬁnition of convergence with $$ε = \(frac{s-t}{2}$$ to produce an $$n$$ with $$s_n > t_n$$.]
2. Prove that if a sequence converges, then its limit is unique. That is, prove that if $$\lim_{n \to \infty }s_n = s$$ and $$\lim_{n \to \infty }s_n = s$$, then $$s = t$$.

### Q3

Prove that if the sequence $$(s_n)$$ is bounded then $$\lim_{n \to \infty }\left (\frac{s_n}{n} \right ) = 0$$.

### Q4

1. Prove that if $$x \neq 1$$, then $1 + x + x^2 +\cdots + x^n = \frac{1 - x^{n+1}}{1-x}$
2. Use (a) to prove that if $$|x| < 1$$, then $$\lim_{n \to \infty }\left ( \sum_{j=0}^{n} x^j \right ) = \frac{1}{1-x}$$

### Q5

Prove $\lim_{n \to \infty }\frac{a_0 + a_1n + a_2n^2 +\cdots + a_kn^k}{b_0 + b_1n + b_2n^2 +\cdots + b_kn^k} = \frac{a_k}{b_k}$

provided $$b_k \neq 0$$. [Notice that since a polynomial only has ﬁnitely many roots, then the denominator will be non-zero when n is suﬃciently large.]

### Q6

Prove that if $$\lim_{n \to \infty }s_n = s$$ and $$\lim_{n \to \infty }(s_n - t_n) = 0$$, then $$\lim_{n \to \infty }t_n = s$$.

### Q7

1. Prove that if $$\lim_{n \to \infty }s_n = s$$ and $$s < t$$, then there exists a real number $$N$$ such that if $$n > N$$ then $$s_n < t$$.
2. Prove that if $$\lim_{n \to \infty }s_n = s$$ and $$r < s$$, then there exists a real number $$M$$ such that if $$n > M$$ then $$r < s_n$$.

### Q8

Suppose $$(s_n)$$ is a sequence of positive numbers such that $$\lim_{n \to \infty }\left ( \frac{s_{n+1}}{s_n} \right ) = L$$

1. Prove that if $$L < 1$$, then $$\lim_{n \to \infty }s_n = 0$$. [Hint: Choose $$R$$ with $$L < R < 1$$. By the previous problem, $$∃\; N$$ such that if $$n > N$$, then $$\frac{s_{n+1}}{s_n} < R$$. Let $$n_0 > N$$ be ﬁxed and show $$s_{n_0+k} < R^ks_{n_0}$$. Conclude that $$\lim_{k \to \infty }s_{n_0+k} = 0$$ and let $$n = n_0 + k$$.]
2. Let $$c$$ be a positive real number. Prove $\lim_{n \to \infty }\left ( \frac{c^n}{n!} \right ) = 0$

### Contributor

• Eugene Boman (Pennsylvania State University) and Robert Rogers (SUNY Fredonia)