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Section 10.4 Geometric series
Objectives
Interpret geometric sequences and series as repeated multiplication leading to an infinite sum.
Derive the geometric series formula and determine its interval of convergence.
Generate further Maclaurin expansions from geometric series.
A geometric sequence is a sequence in which the terms are in a constant common ratio from one to the next.
A geometric series is the infinite sum.
Sum of a geometric series: shift and subtract
Graph the function... it only converges when
\(-1\lt x\lt 1\text{.}\) Verify that this makes sense with the numerical behavior. Takeaway: we have to watch out for where any given power series converges.
Include Maclaurin series for
\(\dfrac{1}{1+x}\text{,}\) \(\ln(1+x)\text{,}\) \(\dfrac{1}{1+x^{2}}\text{,}\) \(\arctan x\)
Madhava series for
\(\dfrac{\pii}{4}\)
