How did Taylor's formula come up with the idea of expressing approximate functions with multiple terms?

How did Taylor's formula come up with the idea of expressing approximate functions with multiple terms?

Because he is really a genius, I asked the university mathematics teacher, the teacher can not answer how to think of

F (x + H) second order Taylor formula
F (x + H) = f (x) + (x - (x + H)) f '(x) = f (x) - f' (x) h, right?

It should be equal to f (x) + HF '(x) + F' '(x) H ^ 2 / 2 + Lagrange or piano remainder

Finding Taylor formula of order n with Lagrange remainder expanded by (x + 1) f (x) = 1 / X

The Taylor formula of order n with Lagrange remainder expanded at XO = - 1 with F (x) = 1 / X is as follows:
1/x=-1-(x+1)-(x+1)^2-(x+1)^3-…… -（x+1)^n+(-1)^(n+1)ξ^(-n-2)(x+1)^(n+1)
Where (- 1) ^ (n + 1) ξ ^ (- n-2) (x + 1) ^ (n + 1) is the Lagrangian remainder, ξ ∈ (- 1, x)
The above answers are for reference only,

There is an apple on the table

there are（many） apples on the table

In triangle ABC, the opposite sides of angle A.B.C are A.B.C and cosa = 2, radical 5 / 5. Tanb = 1 / 3. Find the value of Tanc. If the longest side is 1, find the shortest side

Cosa = 2 radical 5 / 5 Sina = radical 5 / 5 Tana = 1 / 2 tanb = 1 / 3 SINB = 1 / 10 Tanc = - (1 / 2 + 1 / 3) / (1-1 / 2 × 1 / 3) = - 1
The value of Tanc is - 1 sinc = 1 / 2 under root
The shortest edge B = c.sinb/sinc = (radical 5) / 5

Live / do / you / where?
Summer English homework, fourth grade are so difficult, dizzy~

Where do you live?
General questions

The school built a cylindrical flower bed, the inner diameter of the bottom is 3 meters, the outer diameter is 4 meters, and the height is 0.8 meters
1. If the height of filling in the flowerbed is 0.6 m, how many cubic meters of filling is needed in the flowerbed? 2. If tiles are pasted on the outer surface of the flowerbed, at least how many square meters of tiles are needed?

1. V = π (d (inner) / 2) & # 178; H
2. S = π D (outer) H
Do it yourself=

Russia (adjective)_______ Busy (comparative)_______ Draw (past tense)_______ Us (nominative)

Russian busier drew we

Let a, B and C be the internal angles of a triangle, and the equation (SINB Sina) x ^ 2 + (Sina sinc) x ^ 2 + (sinc SINB) = 0 has equal roots, then the angle B ()
A.B＞60°
B.B≥60°
C.B＜60°
D.B≤60°
The equation should be (SINB Sina) x ^ 2 + (Sina sinc) x + (sinc SINB) = 0

D

What are the present participle and past tense of English verbs?

The past tense of regular verbs is composed of the original verb form + - ed. the specific changes are as follows: 1. Directly add - ed at the end of the word, such as: want wanted, work worked, need needed, clean clean; 2. Add - D at the end of the word, such as: like like, live, use used, move