{"id":46079,"date":"2024-04-15T22:07:13","date_gmt":"2024-04-15T22:07:13","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=46079"},"modified":"2024-04-15T22:07:13","modified_gmt":"2024-04-15T22:07:13","slug":"the-region-of-convergence-of-z-transform-of-the-sequence-left-5-over-6-rightnuleft-n-right-left-6-over-5-rightnuleft-n-1-right-must-be","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/the-region-of-convergence-of-z-transform-of-the-sequence-left-5-over-6-rightnuleft-n-right-left-6-over-5-rightnuleft-n-1-right-must-be\/","title":{"rendered":"The region of convergence of z-transform of the sequence $${\\left( {{5 \\over 6}} \\right)^n}u\\left( n \\right) &#8211; {\\left( {{6 \\over 5}} \\right)^n}u\\left( { &#8211; n &#8211; 1} \\right)$$ must be"},"content":{"rendered":"<p>[amp_mcq option1=&#8221;$$\\left| z \\right| < {5 \\over 6}$$\" option2=\"$$\\left| z \\right| > {5 \\over 6}$$&#8221; option3=&#8221;$${5 \\over 6} < \\left| z \\right| < {6 \\over 5}$$\" option4=\"$${6 \\over 5} < \\left| z \\right| < \\infty $$\" correct=\"option3\"]<!--more--><\/p>\n<p>The correct answer is $\\boxed{{5 \\over 6} &lt; \\left| z \\right| &lt; {6 \\over 5}}$.<\/p>\n<p>The region of convergence of the z-transform of a sequence is the set of all values of $z$ for which the z-transform converges. The z-transform of the sequence ${\\left( {{5 \\over 6}} \\right)^n}u\\left( n \\right) &#8211; {\\left( {{6 \\over 5}} \\right)^n}u\\left( { &#8211; n &#8211; 1} \\right)$ is given <div class=\"youtube-subscribe-container\">\r\n        <a href=\"https:\/\/www.youtube.com\/channel\/UCNHT8lW-JmLC68rjBfZhdkg?sub_confirmation=1\" target=\"_blank\" class=\"youtube-subscribe-button\">\r\n            <span class=\"youtube-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 576 512\">\r\n                    <path d=\"M549.7 124.1c-6.3-23.7-24.8-42.3-48.3-48.6C458.8 64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z\"\/>\r\n                <\/svg>\r\n       <div class=\"telegram-channel-container\">\r\n        <a href=\"https:\/\/t.me\/pscnotes2025\" target=\"_blank\" class=\"telegram-channel-button\">\r\n            <span class=\"telegram-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 496 512\">\r\n                    <path fill=\"white\" d=\"M248,8C111,8,0,119,0,256s111,248,248,248s248-111,248-248S385,8,248,8z M362,177L320,367c-3,14-10,18-20,14l-56-41l-27,26 c-3,3-5,5-10,5l4-63L323,196c5-5-1-7-8-3l-98,62l-42-13c-9-3-10-9,2-14l162-63C351,160,365,164,362,177z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Join Our Telegram Channel\r\n        <\/a>\r\n    <\/div>      <\/span>\r\n            Subscribe on YouTube\r\n        <\/a>\r\n    <\/div> by<\/p>\n<p>$$Z\\left{ {\\left( {{5 \\over 6}} \\right)^n}u\\left( n \\right) &#8211; {\\left( {{6 \\over 5}} \\right)^n}u\\left( { &#8211; n &#8211; 1} \\right) \\right} = {1 \\over 1 &#8211; {5 \\over 6}z^{-1}} &#8211; {1 \\over 1 &#8211; {6 \\over 5}z}$$<\/p>\n<p>The z-transform converges if and only if the denominator does not have any poles in the region of interest. The denominator has poles at $z = {6 \\over 5}$ and $z = {5 \\over 6}$. Therefore, the region of convergence is the set of all values of $z$ such that $z \\neq {6 \\over 5}$ and $z \\neq {5 \\over 6}$. This can be written as $${5 \\over 6} &lt; \\left| z \\right| &lt; {6 \\over 5}$$<\/p>\n","protected":false},"excerpt":{"rendered":"[amp_mcq option1=&#8221;$$\\left| z \\right| < {5 \\over 6}$$\" option2=\"$$\\left| z \\right| > {5 \\over 6}$$&#8221; option3=&#8221;$${5 \\over 6} < \\left| z \\right| < Subscribe on YouTube {6 \\over 5}$$\" option4=\"$${6 \\over 5} < \\left| z \\right| < \\infty $$\" correct=\"option3\"]\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[959],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>The region of convergence of z-transform of the sequence $${\\left( {{5 \\over 6}} \\right)^n}u\\left( n \\right) - {\\left( {{6 \\over 5}} \\right)^n}u\\left( { - n - 1} \\right)$$ must be<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/the-region-of-convergence-of-z-transform-of-the-sequence-left-5-over-6-rightnuleft-n-right-left-6-over-5-rightnuleft-n-1-right-must-be\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The region of convergence of z-transform of the sequence $${\\left( {{5 \\over 6}} \\right)^n}u\\left( n \\right) - {\\left( {{6 \\over 5}} \\right)^n}u\\left( { - n - 1} \\right)$$ must be\" \/>\n<meta property=\"og:description\" content=\"[amp_mcq option1=&#8221;$$left| z right| &lt; {5 over 6}$$&quot; option2=&quot;$$left| z right| &gt; {5 over 6}$$&#8221; option3=&#8221;$${5 over 6} &lt; left| z right| &lt; {6 over 5}$$&quot; option4=&quot;$${6 Join Our Telegram Channel Subscribe on YouTube over 5} &lt; left| z right| &lt; infty $$&quot; correct=&quot;option3&quot;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/the-region-of-convergence-of-z-transform-of-the-sequence-left-5-over-6-rightnuleft-n-right-left-6-over-5-rightnuleft-n-1-right-must-be\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-15T22:07:13+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"The region of convergence of z-transform of the sequence $${\\left( {{5 \\over 6}} \\right)^n}u\\left( n \\right) - {\\left( {{6 \\over 5}} \\right)^n}u\\left( { - n - 1} \\right)$$ must be","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/the-region-of-convergence-of-z-transform-of-the-sequence-left-5-over-6-rightnuleft-n-right-left-6-over-5-rightnuleft-n-1-right-must-be\/","og_locale":"en_US","og_type":"article","og_title":"The region of convergence of z-transform of the sequence $${\\left( {{5 \\over 6}} \\right)^n}u\\left( n \\right) - {\\left( {{6 \\over 5}} \\right)^n}u\\left( { - n - 1} \\right)$$ must be","og_description":"[amp_mcq option1=&#8221;$$left| z right| < {5 over 6}$$\" option2=\"$$left| z right| > {5 over 6}$$&#8221; 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