Geometry

Topology

Sequences and Convergence Warmup

         

On Day 1, Zeno is 1,000,000 miles from his goal. Each day, he travels 910\frac{9}{10} of the remaining distance to his goal.

Let dn=d_n = the distance remaining at the end of Day n.n. Is there a day after which Zeno will always be less than 14\frac{1}{4} mile from his goal?

Hint.

d1=1,000,000d_1 = 1,000,000

d2=100,000d_2 = 100,000

d3=10,000d_3 = 10,000

d4=1,000d_4 = 1,000

etc.

On Day 1, Zilo is at home, 1,000,000 miles from his goal. He progresses towards his goal on even days, but needs to return home on odd days to check in.

Specifically on even days, he travels to the closest point to his goal he's reached yet, and then proceeds an additional 910\frac{9}{10} of the remaining distance to his goal. On odd days, he goes back to being 1,000,000 miles from his goal.

Let dn=d_n = the distance remaining at the end of Day n.n. Is there a day after which Zilo will always be closer than 14\frac{1}{4} mile from his goal?

Hint.

d1=1,000,000d_1 = 1,000,000

d2=100,000d_2 = 100,000

d3=1,000,000d_3 = 1,000,000

d4=10,000d_4 = 10,000

d5=1,000,000d_5 = 1,000,000

d6=1,000d_6 = 1,000

etc.

Consider the sequence {(1)n}={1,+1,1,+1,1,+1,}.\{(-1)^n\} = \{-1, +1, -1, +1, -1, +1, \dots\}.

What are the subsequential limit(s) of this sequence?

Note. A subsequence of a sequence {xn}={x1,x2,}\{x_n\}= \{x_1, x_2, \ldots\} is a sequence {xnk}={xn1,xn2,}\{x_{n_k}\}= \{x_{n_1}, x_{n_2}, \ldots\} where n1<n2<n_1 < n_2 < \cdots are natural numbers.

Consider the sequence

{an}={1,122,13,142,15,162,17,182,19,1102,}\{a_n\} = \{1,\frac{1}{2^2}, \frac{1}{3}, \frac{1}{4^2}, \frac{1}{5}, \frac{1}{6^2}, \frac{1}{7}, \frac{1}{8^2}, \frac{1}{9}, \frac{1}{10^2}, \ldots \}

defined by an={1nif n is odd1n2if n is evena_n = \begin{cases} \frac{1}{n} & \textrm{if } n \textrm{ is odd} \\ \frac{1}{n^2} & \textrm{if } n \textrm{ is even} \\ \end{cases}

for each natural number n.n. What is the smallest value of NN such that

an<164 whenever n>N?a_n < \frac{1}{64} \text{ whenever } n >N?

Does the sequence

{an}={5+(1)nn}\{a_n\} = \{5 + \frac{(-1)^n}{n}\}

converge?

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