Certainly! Let's solve the equation [tex]\( t^t = 1 + t + \frac{t^2}{2} + \frac{t^3}{6} \)[/tex].
### Step-by-Step Solution:
1. Consider the Equation:
[tex]\[
t^t = 1 + t + \frac{t^2}{2} + \frac{t^3}{6}
\][/tex]
2. Analyze the Right-Hand Side (RHS):
The RHS is the Taylor series expansion of [tex]\( e^t \)[/tex] (the exponential function [tex]\( e^t \)[/tex] can be expanded as [tex]\( 1 + t + \frac{t^2}{2} + \frac{t^3}{6} + \ldots \)[/tex]), but truncated at the [tex]\( t^3 \)[/tex] term. Hence it approximates [tex]\( e^t \)[/tex] for small values of [tex]\( t \)[/tex].
3. Intuitive Observation:
For small values of [tex]\( t \)[/tex], the Taylor series approximation [tex]\( 1 + t + \frac{t^2}{2} + \frac{t^3}{6} \)[/tex] is close to [tex]\( e^t \)[/tex]. This indicates that [tex]\( t \)[/tex] itself should be a value such that [tex]\( t^t \approx e^t \)[/tex], especially for [tex]\( t \)[/tex] very small.
4. Trial with Simple Values:
Trying [tex]\( t = 1 \)[/tex]:
[tex]\[
1^1 = 1
\][/tex]
[tex]\[
1 + 1 + \frac{1^2}{2} + \frac{1^3}{6} = 1 + 1 + \frac{1}{2} + \frac{1}{6} = 2 + \frac{2}{3} = 2.6667
\][/tex]
[tex]\( 1 \neq 2.6667 \)[/tex]
Trying [tex]\( t = 0 \)[/tex]:
[tex]\[
0^0 \quad \text{(by convention we take } 0^0 = 1 \text{)}
\][/tex]
[tex]\[
1 + 0 + \frac{0^2}{2} + \frac{0^3}{6} = 1
\][/tex]
[tex]\( 1 = 1 \)[/tex]. This is a valid solution.
Now, consider other possible values of [tex]\( t \)[/tex].
5. Approximation Method:
For more precise values, more sophisticated numerical methods or approximations might be necessary, but based on analysis and the form of the functions, we will test a few likely candidates close to zero.
Trying [tex]\( t=0.5 \)[/tex]:
[tex]\[
0.5^{0.5} = \sqrt{0.5} \approx 0.707
\][/tex]
[tex]\[
1 + 0.5 + \frac{0.5^2}{2} + \frac{0.5^3}{6} = 1 + 0.5 + 0.125 + \frac{0.125}{3} \approx 1.6667
\][/tex]
6. Using Numerical Analysis:
To find a more accurate solution, it involves iterative numerical methods, like the Newton-Raphson method. However, since [tex]\( t = 0 \)[/tex] exactly satisfies the equation:
### Conclusion:
By observation and simple calculation, [tex]\( t = 0 \)[/tex] is a clear solution to the equation:
[tex]\[
t^t = 1 + t + \frac{t^2}{2} + \frac{t^3}{6}
\][/tex]
Other values may only be approximations, but [tex]\( t = 0 \)[/tex] is an exact and valid solution.
