Question

Produce the power series for tanx with the term x⁶.*

a) x + (x³/3) + (2x⁵/15) + (17x⁷/315)
b) x + (x³/3) + (x⁵/5) + (x⁷/7)
c) x + (x³/3) + (x⁵/5) + (2x⁷/7)
d) x + (x³/3) + (x⁵/5) + (17x⁷/105)

Answer 1

The power series for tan(x) up to the term with x6 (or more accurately, x7) is given by Option a) which is x + (x3/3) + (2x5/15) + (17x7/315); this matches the known series expansion for tan(x) with only odd powers of x and specific coefficients.

Step-by-step explanation:

To find the power series for tan(x), we look for the expansion that matches the formula given for the series. The term with x6 that the student is referring to would be one degree higher, x7, as power series of tan(x) contain only odd powers of x. The correct option from the given choices should start from x, and then the coefficient of each subsequent term must be such that it corresponds to the series expansion of tan(x).

Looking at the list:

• Option a) gives x + (x3/3) + (2x5/15) + (17x7/315)
• Option b) gives x + (x3/3) + (x5/5) + (x7/7)
• Option c) gives x + (x3/3) + (x5/5) + (2x7/7)
• Option d) gives x + (x3/3) + (x5/5) + (17x7/105)

The power series for tan(x) can be derived using the Maclaurin series for trigonometric functions and, for tan(x), includes only the odd powers of x and has specific coefficients. Among the provided options, the correct series expansion for tan(x) is Option a) because it has the coefficients that match the known power series for tan(x).