Question

What is the general expression for the nth term in the Taylor series at x0 for ln(1 + 3x⁵)?

Answer 1

The general expression for the nth term in the Taylor series at x0 for ln(1 + 3x⁵) is 3x⁵ - (3x⁵)²/2 + (3x⁵)³/3 - (3x⁵)⁴/4 + ...

Step-by-step explanation:

The general expression for the nth term in the Taylor series at x0 for ln(1 + 3x⁵) can be calculated using the formula for the Taylor series expansion of ln(1 + x). The formula is:

Taylor series expansion for ln(1 + x):

1. ln(1 + x) = x - x²/2 + x³/3 - x⁴/4 + ...

By substituting 3x⁵ for x in the formula above, we can find the general expression for ln(1 + 3x⁵):

1. ln(1 + 3x⁵) = 3x⁵ - (3x⁵)²/2 + (3x⁵)³/3 - (3x⁵)⁴/4 + ...

This expression represents the nth term in the Taylor series at x0 for ln(1 + 3x⁵).