The pillar of Infinity

Answer at t = 1 is 1.
Answer at t = 2 is 1.
Answer at t = 3 is 1.
Answer at t = 4 is 1.

If we look carefully all the patterns are of the form $\dfrac{\dfrac{x}{y}}{\dfrac{zx}{zy}}$ , which is 1, so the answer is 0.5 + (1 × 3600) = $\boxed{3600.5}$

An alternate solution is given roughly by $1.27447 \times {10}^{11241}$ . We note that

$\huge \frac { 1 }{ 2 } =\frac { 2(1-1)! }{ 3-{(-1)}^{1}}$

$\huge \frac { { 1 }/{ 4 } }{ { 2 }/{ 8 } } =\frac { 2(2-1)! }{ 3-(-1)^{ 2 } } =\quad 1$

$\huge \frac { \frac { 1 / 4 }{ 4/16 } }{ \frac { 2/8 }{ 8/32 } } =\frac { 2(3-1)! }{ 3-(-1)^{ 3 } } = 1$ .

Continuing this pattern, we may let the $n$ th term of the sequence be given by

$\huge \frac { 2(n-1)! }{ 3-{ (-1) }^{ n } }$ .

Thus the series is given by

$\huge \sum _{ n=1 }^{ 3601 }{\quad \frac { 2(n-1)! }{ 3-{ (-1) }^{ n } } } \cong 1.2744\times { 10 }^{ 11241 }$ .