Euler and History
What city did Euler spend much of his career in?
St. Petersburg
What is the Basel Problem asking you to sum?
1 + 1/4 + 1/9 + 1/16 + ... (reciprocals of perfect squares)
What function did Euler write as an infinitely long polynomial?
sin(x)
True or False: Mathematicians have found a closed form for Σ 1/k³?
What physical disability did Euler develop but continued working through?
Blindness
The Bernoullis showed the sum converged to a value less than what number?
2
What did Euler do to sin(x)/x to connect it to the sum?
Wrote it as an infinite product using its roots
π²/6 ≈ 1.644 — what is significant about the number 1644 in the history of this problem?
1644 is the year the Basel Problem was first posed
Approximately how many papers did Euler publish in his lifetime?
Over 800
What is the answer to the Basel Problem?
π²/6
What technique did Euler use to extract π²/6?
Comparing/matching coefficients of x²
Euler found closed forms for every even power up to what?
The 26th power (Σ 1/k²⁶)
What famous math family had tried and failed to solve the Basel Problem before Euler?
The Bernoullis
What is the probability that two random integers are coprime, and how does it connect to Euler's result?
6/π² ≈ 60.8% (the reciprocal of π²/6)
Why was Euler's original proof considered not fully rigorous?
He assumed what works for finite polynomials also works for infinite ones, which wasn't justified
What does Σ 1/k⁴ equal?
π⁴/90