Final answer:
To find the power series expansion of the function f(t) = 1 + t² about t = 0, use the binomial theorem to expand the function and find the first four nonzero terms and the general term. The first four nonzero terms are 1, t², ⁿC₂t⁴, and the general term is ⁿCₙt²ⁿ.
Step-by-step explanation:
To find the power series expansion of the function f(t) about t = 0, we need to find the first four nonzero terms and the general term. The function f(t) is given as f(t) = 1 + t². We can write the function as f(t) = 1 + t² = (1)(1 + t²).
Now, let's expand the function using the binomial theorem: (a + b)ⁿ = aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + ...
Substituting a = 1 and b = t², we have:
f(t) = (1)(1 + t²) = 1 + (1)(t²) = 1 + t² + ⁿC₂(1)(t²)² + ...
The first four nonzero terms are 1, t², ⁿC₂t⁴, and the general term can be written as ⁿCₙt²ⁿ where ⁿCₙ represents the binomial coefficient.
The student's question is asking to find the first four nonzero terms and the general term for the power series expansion of the function f(t) = 1 / (1 + t²) about t = 0. This falls under the topic of calculus, particularly dealing with series expansions.
To find the power series expansion, we utilize the binomial theorem. Since 1 / (1 + t²) can be written as (1 + t²)⁻¹, applying the binomial theorem gives us:
1st term: 1
2nd term: -t²
3rd term: t⁴
4th term: -t⁶
Thus, the first four nonzero terms in the power series are 1, -t², t⁴, and -t⁶ respectively. The general term for the power series expansion can be represented as (-1)⁹t²n for n ≥ 0, where n is an integer.