Mathematics, 16.01.2021 14:00 koggebless
Determine the above sequence converges or diverges. If the sequence converges determine its limit
Answers: 1
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th partial sum of a series is the sum of its first 
terms. In symbols, if 
denote the 
th term of the original series, the 
th partial sum of this series would be:
.
, exists (should be a finite number.)
.
in the expression for the 
is an alternating series (signs of consecutive terms alternate,) it would be convergent (that is: the partial sum limit 
.
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while the common ratio is 
. Apply the formula for the sum of geometric series to find an expression for 
:
.![\begin{aligned} \lim\limits_{n \to \infty} S_{n} &= \lim\limits_{n \to \infty} \left(-\frac{2}{3} + \frac{2}{3} \cdot {\left(-\frac{1}{2}\right)}^{n}\right) \\ &= -\frac{2}{3} + \frac{2}{3} \cdot \underbrace{\lim\limits_{n \to \infty} \left[{\left(-\frac{1}{2}\right)}^{n} \right] }_{0}= -\frac{2}{3}\end{aligned}}_](/tpl/images/1041/4921/55a8e.png)
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Determine the above sequence converges or diverges. If the sequence converges determine its limit
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