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Lilit [14]
3 years ago
14

Determine whether the series is convergent or divergent. 1 2 3 4 1 8 3 16 1 32 3 64 convergent divergent Correct:

Mathematics
1 answer:
Vanyuwa [196]3 years ago
5 0

Answer:

This series diverges.

Step-by-step explanation:

In order for the series to converge, i.e. \lim_{n \to \infty} a_n =A it must hold that for any small \epsilon>0, there must exist n_0\in \mathbb{N} so that starting from that term of the series all of the following terms satisfy that  |a_n-A|n_0 .

It is obvious that this cannot hold in our case because we have three sub-series of this observed series. One of them is a constant series with a_n=1 , the other is constant with a_n=3 , and the third one has terms that are approaching infinity.

Really, we can write this series like this:

a_n=\begin{cases} 1 \ , \ n=4k+1, k\in \mathbb{N}_0\\ 2^{k}\ , \ n=2k, k\in \mathbb{N}_0\\3\ , \ n=4k+3, k\in \mathbb{N}_0\end{cases}

If we  denote the first series as b_n=1, we will have that \lim_{k \to \infty} b_k=1.

The second series is denoted as c_k=2^k and we have that \lim_{k \to \infty} c_k=+\infty.

The third sub-series d_k=3 is a constant series and it holds that \lim_{k \to \infty} d_k=3.

Since those limits of sub-series are different, we can never find such n_0\\ so that every next term of the entire series is close to one number.

To make an example, if we observe the first sub-series if follows that A must be equal to 1. But if we chose \epsilon =1, all those terms associated with the third sub-series will be out of this interval (A-1, A+1)=(0, 2).

Therefore, the observed series diverges.

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Part (a)

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Since f(-1) = 4, this means we can then say

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To evaluate g(4), we'll follow the same idea as what we did with f(x). However, we'll start at x = 4 and draw a vertical line until we reach the g(x) curve this time.

<h3>Answer:  4</h3>

==========================================================

Part (b)

We use the same idea as part (a)

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<h3>Answer:  6</h3>

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Part (c)

Same idea as the last two parts. We start on the inside and work toward the outside. Keep in mind that g(x) is now the inner function for this part and for part (d) as well.

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<h3>Answer:   5</h3>

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Part (d)

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