vazelinahS

2020-11-08

Consider the following infinite series.

a.Find the first four partial sums${S}_{1},{S}_{2},{S}_{3},$ and ${S}_{4}$ of the series.

b.Find a formula for the nth partial sum${S}_{n}$ of the indinite series.Use this formula to find the next four partial sums ${S}_{5},{S}_{6},{S}_{7}$ and ${S}_{8}$ of the infinite series.

c.Make a conjecture for the value of the series.

$\sum _{k=1}^{\mathrm{\infty}}\frac{2}{(2k-1)(2k+1)}$

a.Find the first four partial sums

b.Find a formula for the nth partial sum

c.Make a conjecture for the value of the series.

Elberte

Skilled2020-11-09Added 95 answers

Step 1.

We have to consider the infinite series:

(a) To find the first four partial sums

b.Find a formula or the nth partial sum

c. Make a conjecture for the value of the series

Step 2

(a) To find the first four partial sums

Step 3

(b) From the partial sums

So, we can write

Now we find

(c) To make a conjecture of the value of the series.

We can also write

On cancelling out the terms we only left wiyh 1, So

Therefore, the value of the series is 1.

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