Are you good with sum?

Algebra Level 5

A new-born bird was trying to take a fish. After seeing the fish, the bird went into a trajectory described by the function:

\(f\left( x \right) =\displaystyle \sum _{ n=1 }^{ 3 }{ \left(\displaystyle \sum _{ k=1 }^{ n }{ \left( 104{ k }^{ 2 }-435k+385 \right) } { x }^{ n-1 } \right) } \)

The fish saw the bird coming and, in an evasive maneuver performed what can be described by the function:

\(g\left( x \right)= { e }^{ i\pi }{ x }^{ 2 }\quad +\quad \left( { \left( \displaystyle \sum _{ n=1 }^{ \infty }{ \frac { n }{ { 2 }^{ n } } } \right) }^{ 2 }!\quad -\quad \displaystyle \sum _{ k=1 }^{ \infty }{ \frac { 1 }{ { 2 }^{ k } } } \right) x\quad -\quad \left\lfloor \frac { 400 }{ \frac { 1 }{ \displaystyle \sum _{ j=1 }^{ \infty }{ \left( -1 \right) ^{ j+1 }\frac { 1 }{ { 2 }^{ j } } } } } \right\rfloor \quad \)

As you might expect, the new-born bird was fooled by the refraction index of the water, as well as the experience of the fish. Consider that they were at the point of maximum approach of both graphs at the same time and that the unit of the graphs is in meters. Let the minimum distance between them be expressed as A/B where A and B are coprime integers What is A+B?

  • You might use Wolfram Alpha to solve equations of 3rd+ degree.
  • You might use Wolfram Alpha to solve minor calculations.
  • You must use 4 decimal digits for your calculations.
  • You must not use any type of calculator to solve the sums \(\sum \)

Important!! In the final step you must use Wolfram Alpha to find the distance. Once you find it, click on the number and choose an approximate fraction with the numerator not higher than 500.

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