This is just the power series expansion for $\cos\frac\pi4.$Umar said:
The formula for finding the sum of infinite series for 1/sqrt(2) is S = a/(1-r), where "a" is the first term in the series (in this case, 1/sqrt(2)) and "r" is the common ratio (in this case, 1/2).
Yes, it can be calculated using the partial sum formula S_n = a(1-r^n)/(1-r), where "n" is the number of terms in the series. However, this method requires an understanding of limits and may not be feasible for larger values of "n".
The limit of the sum of infinite series for 1/sqrt(2) is 2, meaning that as the number of terms in the series approaches infinity, the sum will converge to 2.
The sum of infinite series for 1/sqrt(2) is convergent because the common ratio (1/2) is less than 1, which is a necessary condition for convergence. Additionally, the limit of the series is a finite number (2) rather than approaching infinity, further confirming convergence.
Yes, the sum of infinite series for 1/sqrt(2) can be used in various real-world applications such as in physics, engineering, and finance. It can be used to calculate the total cost of an investment with compound interest, the total distance traveled by an object under constant acceleration, and more.
