YOUR ANSWER Graphical form: simplifies to Text form: (x^2-x-2)*(x^2-x-2)/(x^2+2*x+1)/(x^2-4*x+4) simplifies to 1 Cartoon (animation) form: For tutors: simplify_cartoon( (x^2-x-2)*(x^2-x-2)/(x^2+2*x+1)/(x^2-4*x+4) ) If you have a website, here's a link to this solution. DETAILED EXPLANATION Look at . Reduce similar several occurrences of to It becomes . Look at . Canceled out common factors (x-(-1)),(x-(-1)) Look at . Remove unneeded parentheses around factor It becomes . Look at . Remove extraneous '1' from product It becomes . Look at . Reduce similar several occurrences of to It becomes . Result: Universal Simplifier and Solver

1. Websnapperpro 2 3 5 X 2 Y 3 5 X 2
2. Websnapperpro 2 3 5 X 20
3. Websnapperpro 2 3 5 X 2 3

So, 2 2 would be typed 2^2. X 2 would be typed x^2. (x+5) 2 would be typed (x+5)^2. You can put a fraction in an exponent. X 2/3 should be typed like x^(2/3). With more complicated fractions you have to use parenthesis. For example if you typed x^2+1/x-5, you might think this means 'the quantity 'x-squared plus 1' over the quantity 'x minus 5.

Welcome to Graphical Universal Mathematical Expression Simplifier and Algebra Solver (GUMESS). It solves most middle school algebra equations and simplifies expressions, and it SHOWS ALL WORK. It is free to use. Enter expression to be simplified, or equation to be solved. I will figure out if what you typed is an equation.

1. 3 x 2 = 6 7 x 2 = 14 3 x 3 = 9 7 x 3 = 21 3 x 4 = 12 7 x 4 = 28 3 x 5 = 15 7 x 5 = 35 3 x 6 = 18 7 x 6.
2. Solve 2x = 5; The variable is the letter on the left-hand side (LHS) of the equation. The variable is multiplied by two. I want the x to stand alone on one side of the equation. Since the x is multiplied by 2, I need to divide on both sides by 2, in order to get the variable by itself.
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You must use the '*' (star) symbol for all multiplications!x(x+1) is WRONG! It means a function x ofx-1. x*(x-1) is right. Use the ^ (caret) forexponentiation. x^2 means x squared. x/2y means
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Websnapperpro 2 3 5 X 2 Y 3 5 X 2

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In mathematics, 1 + 2 + 4 + 8 + ⋯ is the infinite series whose terms are the successive powers of two. As a geometric series, it is characterized by its first term, 1, and its common ratio, 2. As a series of real numbers it diverges to infinity, so in the usual sense it has no sum. In a much broader sense, the series is associated with another value besides ∞, namely −1, which is the limit of the series using the 2-adic metric.

Summation[edit]

The partial sums of 1 + 2 + 4 + 8 + ⋯ are 1, 3, 7, 15, …; since these diverge to infinity, so does the series.

${\displaystyle 2^0+2^1+\cdots +2^k=2^{k+1}-1}$

Therefore, any totally regular summation method gives a sum of infinity, including the Cesàro sum and Abel sum.^[1] On the other hand, there is at least one generally useful method that sums 1 + 2 + 4 + 8 + ⋯ to the finite value of −1. The associated power series

${\displaystyle f(x)=1+2x+4x^2+8x^3+\cdots +2^n{x}^n+\cdots =\frac{1}{1-2x}}$

has a radius of convergence around 0 of only 1/2, so it does not converge at x = 1. Nonetheless, the so-defined function f has a unique analytic continuation to the complex plane with the point x = 1/2 deleted, and it is given by the same rule f(x) = 1/1 − 2x. Since f(1) = −1, the original series 1 + 2 + 4 + 8 + ⋯ is said to be summable (E) to −1, and −1 is the (E) sum of the series. (The notation is due to G. H. Hardy in reference to Leonhard Euler's approach to divergent series).^[2]

An almost identical approach (the one taken by Euler himself) is to consider the power series whose coefficients are all 1, i.e.

${\displaystyle 1+y+y^2+y^3+\cdots =\frac{1}{1-y}}$

and plugging in y = 2. These two series are related by the substitution y = 2x.

The fact that (E) summation assigns a finite value to 1 + 2 + 4 + 8 + … shows that the general method is not totally regular. On the other hand, it possesses some other desirable qualities for a summation method, including stability and linearity. These latter two axioms actually force the sum to be −1, since they make the following manipulation valid:

In a useful sense, s = ∞ is a root of the equation s = 1 + 2s. (For example, ∞ is one of the two fixed points of the Möbius transformationz → 1 + 2z on the Riemann sphere). If some summation method is known to return an ordinary number for s, i.e. not ∞, then it is easily determined. In this case s may be subtracted from both sides of the equation, yielding 0 = 1 + s, so s = −1.^[3]

The above manipulation might be called on to produce −1 outside the context of a sufficiently powerful summation procedure. For the most well-known and straightforward sum concepts, including the fundamental convergent one, it is absurd that a series of positive terms could have a negative value. A similar phenomenon occurs with the divergent geometric series 1 − 1 + 1 − 1 + ⋯, where a series of integers appears to have the non-integer sum 1/2. These examples illustrate the potential danger in applying similar arguments to the series implied by such recurring decimals as 0.111… and most notably 0.999…. The arguments are ultimately justified for these convergent series, implying that 0.111… = 1/9 and 0.999… = 1, but the underlying proofs demand careful thinking about the interpretation of endless sums.^[4]

It is also possible to view this series as convergent in a number system different from the real numbers, namely, the 2-adic numbers. As a series of 2-adic numbers this series converges to the same sum, −1, as was derived above by analytic continuation.^[5]

See also[edit]

Websnapperpro 2 3 5 X 20

• Two's complement, a data convention for representing negative numbers where −1 is represented as if it were 1 + 2 + 4 + ⋯ + 2ⁿ⁻¹.

Notes[edit]

1. ^Hardy p. 10
2. ^Hardy pp.8, 10
3. ^The two roots of s = 1 + 2s are briefly touched on by Hardy p. 19.
4. ^Gardiner pp. 93–99; the argument on p. 95 for 1 + 2 + 4 + 8 + ⋯ is slightly different but has the same spirit.
5. ^Koblitz, Neal (1984). p-adic Numbers, p-adic Analysis, and Zeta-Functions. Graduate Texts in Mathematics, vol. 58. Springer-Verlag. pp. chapter I, exercise 16, p. 20. ISBN0-387-96017-1.

References[edit]

• Euler, Leonhard (1760). 'De seriebus divergentibus'. Novi Commentarii academiae scientiarum Petropolitanae. 5: 205–237.
• Gardiner, A. (2002) [1982]. Understanding infinity: the mathematics of infinite processes (Dover ed.). Dover. ISBN0-486-42538-X.
• Hardy, G. H. (1949). Divergent Series. Clarendon Press. LCCQA295 .H29 1967.

Further reading[edit]

• Barbeau, E. J.; Leah, P. J. (May 1976). 'Euler's 1760 paper on divergent series'. Historia Mathematica. 3 (2): 141–160. doi:10.1016/0315-0860(76)90030-6.
• Ferraro, Giovanni (2002). 'Convergence and Formal Manipulation of Series from the Origins of Calculus to About 1730'. Annals of Science. 59: 179–199. doi:10.1080/00033790010028179.
• Kline, Morris (November 1983). 'Euler and Infinite Series'. Mathematics Magazine. 56 (5): 307–314. doi:10.2307/2690371. JSTOR2690371.
• Sandifer, Ed (June 2006). 'Divergent series'(PDF). How Euler Did It. MAA Online.
• Sierpińska, Anna (November 1987). 'Humanities students and epistemological obstacles related to limits'. Educational Studies in Mathematics. 18 (4): 371–396. doi:10.1007/BF00240986. JSTOR3482354.

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