Problems of Taylor formula in Higher Mathematics E (- x ^ 2 / 2) = 1-x ^ 2 / 2 + 1 / 2! (- x ^ 2 / 2) ^ 2 + O (x ^ 4), why is the infinitesimal of the peyano remainder o (x ^ 4) written like this? Instead of O ((- x ^ 2 / 2) ^ 2) = O (x ^ 4 / 4), I don't understand why?

Similarly, generally only write the coefficient of 1 (you can understand it as an order of magnitude, general mathematicians most want to know what order of magnitude a number is), O (x ^ 4) means x ^ 4 high-order infinitesimal, that is, LIM (O (x ^ 4) / (x ^ 4)) = 0. In other words, O (x ^ 4) is smaller than x ^ 4, which may be x ^ 5, x ^ 6

Find the first n terms and Sn of sequence (2 + 1 / 4), (4 + 1 / 16), (6 + 1 / 64), (2n + 1 / 4N)

=(2+4+.+2n)+(1/4+1/16+...+1/4^n)
=n(n+1)+[1/4^(n+1)-1/4]/(1/4-1)
=n(n+1)-(1/4^n-1)/3

RELATED INFORMATIONS

1. Find the first n terms and Sn of sequence 1 / 2 ^ 2 + 4,1 / 4 ^ 2 + 8,1 / 6 ^ 2 + 12.1 / (2n) ^ 2 + 4N
2. （2-3×5-1）+（4-3×5-2）+… （2n-3×5-n）=______ ．
3. The first four terms of known sequence are 2 / 3,3 / 4,5 / 6. Write a general term formula
4. Is 1 - 8 4 100 - 9 a sequence and why
5. The sum problem of sequence an = 3 / (2n-11) n belongs to positive integer, and the sum of the first n terms of sequence {an} is Sn, so what is the minimum value of n with Sn ＞ 0 Summation of sequence An = 3 / (2n-11) n is a positive integer, and the sum of the first n terms of the sequence {an} is Sn, so what is the minimum value of n with Sn > 0?
6. Given that an = 32n − 11 (n ∈ n *), the sum of the first n terms of the sequence {an} is Sn, then the minimum value of n with Sn > 0 is () A. 10B. 11C. 12D. 13
7. In the arithmetic sequence an, a (n + 1) = 2n + 1, then Sn = (1 / A1A2) + (1 / a2a3) （1/a99a100）=
8. If an = 2n (n ∈ n *), then A1A2 + a2a3 + a3a4 + +anan+1=
9. Let the sequence an satisfy a1 + 3a2 + 3 & # 178; A3 + +3 ^ n-1 (an) = n / 3, find the general term formula of sequence an
10. Given the sequence an satisfies a1 + 3a2 + 3 & # 178; A3 +... + 3 (n-1) an = n & # 178;, find the general term formula of an Is n-1 power of 3
11. On the criterion of limit existence in Higher Mathematics Let x > 0 have a (x)
12. There are several ways to prove the existence of limit in higher number problem. Besides monotone bounded criterion, are there any other ways to prove the existence of limit? Thank you!
13. Solving problems with two important limits lim 2x+3 ( ——— ) x →∞ 2x+1
14. On the limit of higher numbers When x → 0, (1 + x ^ 2) ^ (1 / 3) - 1 is equivalent to (1 / 3) x ^ 2, that is, the limit of the first formula divided by the second formula when x → 0 is equal to 1. How can this be calculated? Please write down the calculation process. Thank you very much
15. What is the limit of the sum of the x power of E and the negative x power of E? Can infinity and infinitesimal be added to find the limit?
16. Two questions about the limit of higher number 1, 1. X tends to find the limit under infinite root ((2x ^ 2) + 2x + 1) / cubic root ((8x ^ 3) - (x ^ 2) + 1) 2. X tends to infinity (((3-4x) ^ 6) ((2x-1) ^ 19)) / (5x + 1) ^ 25 to find the limit
17. Failed to insert picture - = find Lim {3 / (1 ^ 2 * 2 ^ 2) + 5 / (2 ^ 2 times 3 ^ 2) + +(2n+1)/n^2*(n+1)^2}
18. A problem of finding limit in higher numbers~ Find the limit of LIM x → 1 (1-x) Tan π X / 2?
19. High number for limit! Lim x tends to 0 (SiNx / x) ^ 1 / 2
20. Two exercises on derivation of advanced numbers 1. Find the derivative dy / DX of the function y = f (x) determined by the following parametric equation (1)x=2t,y=t^2 (2)x=te^-t,y=e^t 2. Use the logarithmic derivative method to find the derivatives of the following functions (1)y=x^x Please explain the above exercises in detail, thank you~