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vlada-n [284]
3 years ago
10

Find the sum of the first 41 terms of the following series, to the nearest integer. 2, 8,14,...

Mathematics
1 answer:
Shtirlitz [24]3 years ago
3 0

Answer:

5002

Step-by-step explanation:

The given series is an arithmetic progression with

a = the first term

d = common difference

n = number of terms

S = the sum of the nth term

Recall that

S = n/2(2a + (n-1)d)

As such,

a = 2

d = 8 -2 = 14 - 8 = 6

n = 41

Hence

S = 41/2(2*2 + (41 - 1)6)

= 41/2 ( 4+ 240)

= 41 * 244/2

= 5002

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Determine whether the series is convergent or divergent. 1 2 3 4 1 8 3 16 1 32 3 64 convergent divergent Correct:
Vanyuwa [196]

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.

5 0
4 years ago
The passage is written from the point of view of a
kow [346]

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According to the statement

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So, For this purpose, we know that the

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Learn more about passage here

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7 0
2 years ago
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tankabanditka [31]

Answer:

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Step-by-step explanation:

3/4 hour is equal to 45 minutes. Add 30 minutes to that and you have 1 hour and 15 minutes. You will then add the final hour to that to get 2 hours and 15 minutes. Subtract that from 4:15 and you get 2:00.

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Naily [24]
Okay !! what’s the question ?
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LekaFEV [45]

Answer:

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