The given series is convergent and the first term is 4 and the common ratio is -0.5 and the sum of the series is 8/3 .
The given series is 4 - 4/2+ 4/4 - 4/8 +...
We can see that the first term of the geometric series is 4
Hence a = 4
The second term is -4/2 and the third term is 4/4
Hence r = 4 ÷ (-4/2) = -4/2 ÷ 4/4 = - 1/2 or -0.5
Since the common ratio is less than 1 we can say that the series is convergent.
Now for a convergent series we know that the sum of the terms of the series is equal to
S = a / (1-r)
or, S = 4 / (1-(-0.5))
or, S = 4 / 1.5
or, S = 8/3
Hence the sum of the convergent series is 8/3 .
Both the ratio test and the root test are based on a comparison with a geometric series and hence function in comparable scenarios.
In fact, if the ratio test works (that is, if the limit exists and is not equal to 1), then so does the root test; the contrary is not true. The root test is thus more broad, although in practise, determining the limit for regularly encountered kinds of series is frequently challenging.
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