Final answer:
The closed form expression for series b) is likely -ln(1-x), derivative of the Taylor series for ln(1-x), while for c) it might relate to the exponential function e^x with alternating signs. Series a) and d) are not clearly identifiable without additional context.
Step-by-step explanation:
The question asks to find the closed form expressions for given infinite series expansions. To tackle these, we look for patterns in the exponents and the coefficients that suggest a known series or formula could be used, such as the binomial theorem. For each sequence, we can try to express it as a derivative or an integral of a known series, or relate it to a common function's Taylor series expansion.
For part b), the series resembles the Taylor expansion of the natural logarithm, ln(1-x), which suggests that the closed form could be -ln(1-x). For part c), the series is related to the Taylor series of the exponential function, so it might represent the exponential function e^x, but with alternating signs and factorial denominators. To find the exact closed forms for all the series provided, further analysis of the patterns and manipulations of the expressions are needed. Unfortunately, without additional context or a clear pattern, the closed form for series a) and d) are not immediately identifiable from the information given.