# Expand and evaluate the series n=5 ai=2×i^2+3 (2023)

23.05.2023the following question

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Mathematics

Expand and evaluate the series n=5 ai=2×i^2+3

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Mathematics

5. evaluate the series 50 + 10 + 2 + . . 62 the series diverges; it does not have a sum. 62.5

62.5

Step-by-step explanation:

By dividing 50 by 10 and 10 by 2 we can find that r, or the common ratio is 1/5, meaning that this series converges, so it is solvable. The formula for a convergent series is . By plugging in 1/5 for r and 50 for we can find that the sum of this infinite series is 62.5

Mathematics

the series has 8 terms.evaluate the series 1/2,3/2,5/2,…,15/2 you

This is an arithmetic sequence, not a series, (series are infinite, sequences are finite)because each term is a constant difference from the term preceding it, called the common difference.

Technically the sum of any arithmetic sequence is:

s(n)=(2an+dn^2-dn)/2

However the above equation is derived from the fact that the sum of an arithmetic sequence is the average of the first and last terms times the number of terms in the sequence.

All we really have to "solve" for then is the number of terms in the sequence...the rule for the sequence of an arithmetic sequence is:

a(n)=a+d(n-1), a=first term, d=common difference, n=term number. Since we know a=1/2 and d=1 we can solve for n...

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

a(n)=n-1/2, now we just solve for n when a(n)=15/2

15/2=n-1/2

n=16/2

n=8 so there are seven terms, now we can say:

s(8)=8(1/2+15/2)/2

s(8)=32

Mathematics

The first five terms of a sequence are 2, 5/3, 3/2,7/5, and 4/3. Which of the following can be used to evaluate the series

D.

I JUST TOOK THE TEST, PICTURE ATTACHED.

Mathematics

Evaluate the series 4-2 1-0.5 0.25 to s10. round to the nearest tenth. a. 5.50 b.-2.75 c.2.75 d. 2.66

This is a geometric series, so we will be using the sum to n terms of a geometric series given by:

, where a is the first term, r is the common ratio, and n is the number of terms.

Now, we know the first term is 4.
We know the ratio is
And we are finding to the 10th term.

Plugging everything into the equation:

After this, we get: 2.6640625...≈ 2.66 (ie D)

Mathematics

Evaluate the series 4 – 2 + 1 – 0.5 + 0.25 to s10. round to the nearest hundredth.

The initial term of this geometric series is 4, and the common ratio is -1/2. The sum is given by

The sum is approximately 2.66.

Mathematics

Evaluate the series 4 – 2 + 1 – 0.5 + 0.25 to s10. round to the nearest hundredth.

The initial term of this geometric series is 4, and the common ratio is -1/2. The sum is given by

The sum is approximately 2.66.

Mathematics

We can use this power series to approximate the constant  . a) First, evaluate arctan(1) . (You do not need the series to evaluate it.) b) Use your answer from part (a) and the power series above to find a series representation for  . (The answer will be just a series – not a power series.) c) Verify that the series you found in part (b) converges. d) Use your convergent series from part (b) to approximate  with |error| 0.5. e) How many terms would you need to approximate  with |error| 0.001?

(a)

(b)

(c)

Therefore if you sum any three terms of it you get the desired accuracy.

(d)

If you sum 1999 terms you get the desired accuracy.

Step-by-step explanation:

From the information given we know that

(a)

For that you need to find and angle such that = 1, remember that

therefore

(b)

Then you just multiply by 4 and get that

(c)

Using the alternating series test, since the sequence is decreasing and its limit tends to 0 when n tends to infinity the series is convergent.

(d)

Using the estimation theorem of alternating series we know that if denotes the partial sum of the series then

Therefore we are looking for an such that

we just have to solve that inequality, when you solve that inequality you get that

Therefore if you sum any three terms of it you get the desired accuracy.

(e) For this part you need to solve the following inequality

When you solve that inequality you get that

so, if you sum 1999 terms you get the desired accuracy.

Mathematics

For the following telescoping series, find a formula for the nth term of the sequence of partial sums S n . Then evaluate lim n → [infinity] S n to obtain the value of the series or state that the series diverges. ∑ [infinity] k = 1 = 10 ( 5 k − 1 ) ( 5 k + 4 )

I'm guessing the sum is supposed to be

Split the summand into partial fractions:

If , then

If , then

This means

Consider the th partial sum of the series:

The sum telescopes so that

and as , the second term vanishes and leaves us with

Mathematics

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate Lim Sn to obtain the value of the series or state that the series diverges. n→[infinity] [infinity] Σ (4/√k+5 ) - 4/ √ k+6) k=1

Looks like the series is

This series has n-th partial sum

(where is used as a placeholder for the summand)

In each grouped term, the last term is annihilated by the first term of the next group; that is, for instance,

Ultimately, all the middle terms will vanish and we're left with

As , the last term converges to 0 and we're left with

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