Answer:
a) -2/5
b) No sum
Explanation:
The sum will only exist if the magnitude of the base is less than 1. For first term a1 and common ratio r, the infinite sum is ...
S = a1/(1 -r)
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a)
The first term is -2/3, and the common ratio is -2/3. The infinite sum is ...
S = (-2/3)/(1 -(-2/3)) = (-2/3)/(5/3) = -2/5
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b)
The common ratio (-4) has a magnitude larger than 1. There is No Sum.
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Additional comment
The "infinite sum" of the first series can be calculated by most calculators using fewer than 100 terms. The 100th term will have a value on the order of 2.5×10^-18, so will be beyond the ability of most calculators with 16-digit precision to properly add it to the sum. The attachment shows the output from one such calculator.
As we see from the above sum formula, for rational values of a1 and r, the resulting sum will be a rational number. Often, there will be little error by having a calculator compute the "exact" value of the infinite sum using a suitable finite-length series. Some calculators may be more trustworthy than others in this regard. YMMV