By Ma J. M., Qiu R. F.

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**Extra resources for 3-Manifold Containing Separating Incompressible Surfaces of All Positive Genera**

**Example text**

Use the method of infinite series to solve the following differential equations: + 2y = 0 γ ^ = x 40 NEWTON'S CALCULUS (PART 2) 3. Use the fact that y = sinrr satisfies the differential equation y" = — y to show that 9 x3 x5 x7 _x . . sinx = a : _ _ + _ _ _ + 4. Use the fact that y = cosx satisfies the differential equation y" = —y to show that 2 A 8 G _ _x _x _ _X + _x . . cosx = 1 + 5. Use the fact that y = ex satisfies the differential equation y' = y to show that 7 ΧΔ ■} X° 4 X* Ϊ ! + 2 !

Use Equation (12) through the x 1 1 term and a calculator (with eight decimal places at least) to estimate 4[tan _ 1 (l/2) + t a n _ 1 ( l / 3 ) ] . Compare the resulting estimate of π with its actual value. 3. Use the trigonometric formula t&n(A + B) = tan A + tan B 1 — tan A tan B to prove that ττ/4 = 4 t a n _ 1 ( l / 5 ) - t a n _ 1 ( l / 2 3 9 ) . Use Equation (12) through the x 7 term and a calculator (with eight decimal places at least) to estimate 4[4tan _ 1 (l/5) — tan _ 1 (l/239)]. Compare the resulting estimate of π with its actual value.

Step 3: In Fo(x, to) set to = 2 + ti to obtain 3x 2 + 2x(2 + ti) + (2 + i i ) 2 + x - (2 + ii) - 2 = 0, which simplifies to 3ii + t\ + 5x + 2*ix + 3x 2 = 0. Fi(x,ti): Iteration # 1 Step 1: Ignoring all x" with n > 1 in ί \ ( χ , * ι ) , we get x 1 level: 3c^ + 5 = 0. Step 2: cx = - - . Step 3: In F\{x,t\) set fi = - ( 5 x / 3 ) + *2 to obtain \ OX 3(-y+i 2 / OX ) + {-γ+ί 2 ) \ . „ . n dX I \ which simplifies to „, * F 2 (x, t 2 ): 9 4* 2 x 22x 2 3i2 + t\ - -f- + — Iteration # 2 Step 1: Ignoring all x" with n > 2 in F2(x,*2)> we get 22 3c2 + — = 0.