Question Number 217383 by peter frank last updated on 12/Mar/25 $$\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{intergral} \\ $$$$\int_{\mathrm{1}} ^{\infty} \frac{\mathrm{1}}{\mathrm{x}^{\mathrm{p}} }\mathrm{dx}\:\:\mathrm{converges}\:\mathrm{for}\:\mathrm{p}>\mathrm{1} \\ $$$$\:\mathrm{find}\:\mathrm{it}\:\mathrm{value} \\ $$ Answered by mr W last…
Question Number 217377 by ArshadS last updated on 12/Mar/25 $$\mathrm{Let}\:\mathrm{x},\mathrm{y},\mathrm{z}\:\mathrm{be}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{satisfying}\:\mathrm{the}\:\mathrm{equations} \\ $$$$\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}=\:\mathrm{7} \\ $$$$\mathrm{xy}\:+\:\mathrm{yz}\:+\:\mathrm{zx}=\mathrm{10} \\ $$$$\mathrm{xyz}=\mathrm{6} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{y}^{\mathrm{3}} \:+\:\mathrm{z}^{\mathrm{3}} \\ $$ Answered…
Question Number 217378 by MrGaster last updated on 12/Mar/25 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Whether}\:\mathrm{the}\:\mathrm{following}\: \\ $$$$\mathrm{seriesconverge}: \\ $$$$\begin{array}{|c|}{\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}} \left[\mathrm{1}+{n}\:\mathrm{ln}\left(\mathrm{1}−\frac{\mathrm{2}}{\mathrm{2}{n}+\mathrm{1}}\right)\right]}\\\hline\end{array} \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 217356 by SciMaths last updated on 11/Mar/25 Answered by profcedricjunior last updated on 11/Mar/25 $$\boldsymbol{{i}}=\int_{\mathrm{1}} ^{\mathrm{2}} \int_{\boldsymbol{{y}}} ^{\boldsymbol{{y}}^{\mathrm{2}} } \int_{\mathrm{0}} ^{\boldsymbol{{ln}}\left(\boldsymbol{{y}}+\boldsymbol{{z}}\right)} \boldsymbol{{e}}^{\boldsymbol{{x}}} \boldsymbol{{dxdydz}}…
Question Number 217358 by mnjuly1970 last updated on 11/Mar/25 Answered by mr W last updated on 12/Mar/25 $${let}\:\Delta={area}\:{of}\:{triangle}\:\Delta{ABC} \\ $$$${we}\:{have} \\ $$$$\Delta=\frac{\sqrt{\mathrm{2}\left({a}^{\mathrm{2}} {b}^{\mathrm{2}} +{b}^{\mathrm{2}} {c}^{\mathrm{2}}…
Question Number 217359 by ArshadS last updated on 11/Mar/25 $$\mathrm{Find}\:\mathrm{all}\:\mathrm{three}-\mathrm{digit}\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that}\:\mathrm{when}\:\mathrm{the}\:\mathrm{number}\:\mathrm{is} \\ $$$$\mathrm{divided}\:\mathrm{by}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{its}\:\mathrm{digits}\:\mathrm{the}\:\mathrm{quotient}\:\mathrm{is}\:\mathrm{7}\:\mathrm{and}\:\mathrm{the}\: \\ $$$$\mathrm{remainder}\:\mathrm{is}\:\mathrm{5}. \\ $$ Answered by Rasheed.Sindhi last updated on 12/Mar/25 $$\mathrm{100}{h}+\mathrm{10}{t}+{u}=\mathrm{7}\left({h}+{t}+{u}\right)+\mathrm{5} \\…
Question Number 217352 by issac last updated on 11/Mar/25 $$\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{k}}\:\mathrm{is}\:\mathrm{Divergence}. \\ $$$$\underset{{p}\in\mathbb{P}} {\sum}\:\frac{\mathrm{1}}{{p}}=\:??\:\:\:\mathbb{P}\:\mathrm{is}\:\mathrm{set}\:\mathrm{of}\:\mathrm{prime}\:\mathrm{number} \\ $$$$\underset{{p}\in\mathbb{P}} {\sum}\:\frac{\mathrm{1}}{{p}}=\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{5}}+\frac{\mathrm{1}}{\mathrm{7}}+\frac{\mathrm{1}}{\mathrm{11}}+\frac{\mathrm{1}}{\mathrm{13}}+\frac{\mathrm{1}}{\mathrm{17}}+\frac{\mathrm{1}}{\mathrm{19}}+….. \\ $$$$\mathrm{and}\:\:\mathrm{does}\:\underset{{k}\in\mathbb{N}\backslash\left\{\mathbb{P}\right\}} {\sum}\frac{\mathrm{1}}{{k}}\:\mathrm{is}\:\mathrm{Divergence}..?? \\ $$$$\mathbb{N}\backslash\left\{\mathbb{P}\right\}\:\mathrm{is} \\ $$$$\mathrm{Set}\:\mathrm{of}\:\mathrm{Natural}\:\mathrm{number}−\mathrm{Prime}\:\mathrm{Number}…
Question Number 217326 by Rasheed.Sindhi last updated on 10/Mar/25 $${Find}\:{three}\:{consecutive}\:{integers}\: \\ $$$${such}\:{that}\:{the}\:{sum}\:{of}\:{their}\:{squares} \\ $$$$\:{is}\:\mathrm{50}. \\ $$ Answered by Hanuda354 last updated on 10/Mar/25 $$\mathrm{Let}\:{n},\:{n}+\mathrm{1}\:\:\mathrm{and}\:\:{n}+\mathrm{2}\:\:\mathrm{are}\:\:\mathrm{required}\:\:\mathrm{numbers}. \\…
Question Number 217321 by ArshadS last updated on 10/Mar/25 $$\mathrm{Find}\:\mathrm{three}\:\mathrm{consecutive}\:\mathrm{integers}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{first}\:\mathrm{two}\:\mathrm{is}\:\mathrm{16}\:\mathrm{more}\:\mathrm{than}\:\mathrm{twice}\:\mathrm{the}\:\mathrm{third}\:\mathrm{integer}.\:\mathrm{Provide}\:\mathrm{all}\: \\ $$$$\mathrm{possible}\:\mathrm{solutions}. \\ $$ Answered by Rasheed.Sindhi last updated on 10/Mar/25 $${let}\:{x}−\mathrm{1},{x},{x}+\mathrm{1}\:{are}\:{required}\:{numbers} \\…
Question Number 217300 by peter frank last updated on 09/Mar/25 Answered by Marzuk last updated on 09/Mar/25 $$\:\:\:\:\:\:\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:−\:\mathrm{2}{a}\frac{{dy}}{{dx}}\:+\:\left({a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \right){y}\:=\:\mathrm{0} \\ $$$${or},\frac{{dy}'}{{dx}}\:−\:\mathrm{2}{a}\frac{{d}}{{dx}}\:\left[{e}^{{ax}} {sin}\left({bx}\right)\:\right]\:+\:{a}^{\mathrm{2}}…