Problem 45770. Approximate functions 1 (★★★)

Created by Anirban Pal

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<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 208.3px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 104.15px; transform-origin: 407px 104.15px; vertical-align: baseline; "><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 201.483px 10.5px; transform-origin: 201.483px 10.5px; unicode-bidi: normal; "><span style="">Compute the following sum for any given x and positive integer n.</span></span></div><div style="block-size: 46px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 23px; text-align: center; transform-origin: 384px 23px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><img class="imageNode" style="vertical-align: baseline" src="data:image/gif;base64,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" data-image-state="image-loaded"></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 277.65px 10.5px; transform-origin: 277.65px 10.5px; unicode-bidi: normal; "><span style="">Here m! denotes the factorial of m, which the product of all natural numbers upto m. Thus,</span></span></div><div style="background-color: rgb(247, 247, 247); block-size: 61.3px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 30.65px; transform-origin: 404px 30.65px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 50.6px 8.5px; transform-origin: 50.6px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; perspective-origin: 4.21667px 8.5px; transform-origin: 4.21667px 8.5px; ">2</span><span style="border-block-end-color: rgb(178, 140, 0); border-block-start-color: rgb(178, 140, 0); border-bottom-color: rgb(178, 140, 0); border-inline-end-color: rgb(178, 140, 0); border-inline-start-color: rgb(178, 140, 0); border-left-color: rgb(178, 140, 0); border-right-color: rgb(178, 140, 0); border-top-color: rgb(178, 140, 0); caret-color: rgb(178, 140, 0); color: rgb(178, 140, 0); column-rule-color: rgb(178, 140, 0); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(178, 140, 0); perspective-origin: 46.3833px 8.5px; text-decoration: none; text-decoration-color: rgb(178, 140, 0); text-emphasis-color: rgb(178, 140, 0); transform-origin: 46.3833px 8.5px; ">! = 1x2 = 2</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 59.0333px 8.5px; transform-origin: 59.0333px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; perspective-origin: 4.21667px 8.5px; transform-origin: 4.21667px 8.5px; ">3</span><span style="border-block-end-color: rgb(178, 140, 0); border-block-start-color: rgb(178, 140, 0); border-bottom-color: rgb(178, 140, 0); border-inline-end-color: rgb(178, 140, 0); border-inline-start-color: rgb(178, 140, 0); border-left-color: rgb(178, 140, 0); border-right-color: rgb(178, 140, 0); border-top-color: rgb(178, 140, 0); caret-color: rgb(178, 140, 0); color: rgb(178, 140, 0); column-rule-color: rgb(178, 140, 0); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(178, 140, 0); perspective-origin: 54.8167px 8.5px; text-decoration: none; text-decoration-color: rgb(178, 140, 0); text-emphasis-color: rgb(178, 140, 0); transform-origin: 54.8167px 8.5px; ">! = 1x2x3 = 6</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 84.3333px 8.5px; transform-origin: 84.3333px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; perspective-origin: 4.21667px 8.5px; transform-origin: 4.21667px 8.5px; ">5</span><span style="border-block-end-color: rgb(178, 140, 0); border-block-start-color: rgb(178, 140, 0); border-bottom-color: rgb(178, 140, 0); border-inline-end-color: rgb(178, 140, 0); border-inline-start-color: rgb(178, 140, 0); border-left-color: rgb(178, 140, 0); border-right-color: rgb(178, 140, 0); border-top-color: rgb(178, 140, 0); caret-color: rgb(178, 140, 0); color: rgb(178, 140, 0); column-rule-color: rgb(178, 140, 0); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(178, 140, 0); perspective-origin: 80.1167px 8.5px; text-decoration: none; text-decoration-color: rgb(178, 140, 0); text-emphasis-color: rgb(178, 140, 0); transform-origin: 80.1167px 8.5px; ">! = 1x2x3x4x5 = 120</span></span></div></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 84.4167px 10.5px; transform-origin: 84.4167px 10.5px; unicode-bidi: normal; "><span style="">Avoid using for/while loops.</span></span></div></div></div>

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factorial function sum

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Problem 45770. Approximate functions 1 (★★★)

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