Đặt A=1+1/4+1/42+1/43+....+1/42000
\(4A=4\left(1+\frac{1}{4}+...+\frac{1}{4^{2000}}\right)\)
\(4A=4+1+...+\frac{1}{4^{1999}}\)
\(4A-A=\left(4+1+...+\frac{1}{4^{1999}}\right)-\left(1+\frac{1}{4}+...+\frac{1}{4^{2000}}\right)\)
\(3A=4-\frac{1}{4^{2000}}\)
\(A=\frac{4-\frac{1}{4^{2000}}}{3}\)
Đặt tổng trên là A
Ta có: A=1+1/4+1/42+1/43+...+1/42000 (1)
1/4.A= 1/4+1/42+1/43+1/44+...+1/42001(2)
Lấy (2) trừ (1), ta có:
-3/4.A=1/42001-1
A=(1/42001-1):(-3/4)
\(\text{Đặt }\)\(A=1+\frac{1}{4}+\frac{1}{4^2}+\frac{1}{4^3}+.....+\frac{1}{4^{2000}}\)
\(Khi\text{đ}\text{ó}4\text{A}=4+1+\frac{1}{4}+.....+\frac{1}{4^{2001}}\)
\(\Rightarrow4A-A=4+\frac{1}{4^{2001}}-\frac{1}{4^{2000}}\)
