Răspuns:
S-a demonstrat
Explicație pas cu pas:
Fie a = 1/(1 x 3) + 1/(3 x 5) + ... + 1/((2n - 1) x (2n + 1))
2a = 2/(1 x 3) + 2/(3 x 5) + ... + 2/((2n - 1) x (2n + 1))
Folosim proprietatea: k/(n x (n+k)) = 1/n - 1/(n+k)
2a = 1/1 - 1/3 + 1/3 - 1/5 + ... +1/(2n - 1) - 1/(2n + 1)
Anulam termenii inutili si obtinem:
2a = 1/1 - 1/(2n + 1)
2a = (2n + 1) / (2n + 1) - 1 / (2n + 1)
2a = (2n + 1 - 1) / (2n + 1)
2a = 2n / (2n + 1)
a = n / (2n + 1)
a = 1/(1 x 3) + 1/(3 x 5) + ... + 1/((2n - 1) x (2n + 1))
deci
1/(1 x 3) + 1/(3 x 5) + ... + 1/((2n - 1) x (2n + 1)) = n/ (2n + 1)
